3.6 the chain rule - dr. travers page of...
TRANSCRIPT
![Page 1: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/1.jpg)
§ 3.6 The Chain Rule
![Page 2: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/2.jpg)
The Rule
Ruledydx
=dydu· du
dx
or
Ruleddx
(f ◦ g)(x) =ddx
f (g(x)) = f ′(g(x)) · g′(x)
or
Ruleddx
f (u) = f ′(u)dudx
![Page 3: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/3.jpg)
The Rule
Ruledydx
=dydu· du
dx
or
Ruleddx
(f ◦ g)(x) =ddx
f (g(x)) = f ′(g(x)) · g′(x)
or
Ruleddx
f (u) = f ′(u)dudx
![Page 4: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/4.jpg)
The Rule
Ruledydx
=dydu· du
dx
or
Ruleddx
(f ◦ g)(x) =ddx
f (g(x)) = f ′(g(x)) · g′(x)
or
Ruleddx
f (u) = f ′(u)dudx
![Page 5: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/5.jpg)
What This Means
When we have a composite function, we first identify the inside andoutside functions.
Then we find the derivative of the outside function but leave theinside function as it’s variable.
Then we multiply this result by the derivative of the inside function.
![Page 6: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/6.jpg)
What This Means
When we have a composite function, we first identify the inside andoutside functions.
Then we find the derivative of the outside function but leave theinside function as it’s variable.
Then we multiply this result by the derivative of the inside function.
![Page 7: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/7.jpg)
What This Means
When we have a composite function, we first identify the inside andoutside functions.
Then we find the derivative of the outside function but leave theinside function as it’s variable.
Then we multiply this result by the derivative of the inside function.
![Page 8: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/8.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 9: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/9.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?
u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 10: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/10.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1
What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 11: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/11.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?
u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 12: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/12.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 13: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/13.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?
y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 14: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/14.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 15: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/15.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?
y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 16: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/16.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 17: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/17.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx
= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 18: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/18.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 19: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/19.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 20: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/20.jpg)
Example 1
Example
Find y′ if y = (4x2 + 1)7.
What is the inside function?u = 4x2 + 1What is the derivative?u′ = 8x
What is the outside function?y = u7
What is the derivative?y′ = 7u6
Putting this together, we have
dydx
=dydu· du
dx= 7u6 · 8x
= 7(4x2 + 1)6 · 8x
= 56x(4x2 + 1)6
![Page 21: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/21.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 22: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/22.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?
u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 23: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/23.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2
What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 24: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/24.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?
u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 25: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/25.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 26: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/26.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?
y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 27: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/27.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
u
What is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 28: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/28.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?
y′ = 12√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 29: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/29.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 30: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/30.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 31: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/31.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 32: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/32.jpg)
Example 2
Example
Find y′ if y =√
3x2 + 5x− 2.
What is the inside function?u = 3x2 + 5x− 2What is the derivative?u′ = 6x + 5
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
uPutting this together, we have
dydx
=dydu· du
dx
=1
2√
u· (6x + 5)
=6x + 5
2√
3x2 + 5x− 2
![Page 33: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/33.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 34: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/34.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?
u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 35: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/35.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 36: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/36.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?
u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 37: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/37.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 38: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/38.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?
y = 1u
What is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 39: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/39.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
u
What is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 40: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/40.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?
y′ = − 1u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 41: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/41.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 42: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/42.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 43: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/43.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 44: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/44.jpg)
Example 3
Example
Find y′ if y = 1x2+x4 .
What is the inside function?u = x2 + x4
What is the derivative?u′ = 2x + 4x3
What is the outside function?y = 1
uWhat is the derivative?y′ = − 1
u2
Putting this together, we have
dydx
=dydu· du
dx
= − 1u2 · (2x + 4x3)
= − 2x + 4x3
(x2 + x4)2
![Page 45: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/45.jpg)
Example 4
Example
Find y′ if y = sin4 2x
What is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 46: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/46.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?
u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 47: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/47.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2x
What is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 48: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/48.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?
u′ = cos 2x · ddx 2x
u′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 49: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/49.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x
· ddx 2x
u′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 50: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/50.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2x
u′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 51: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/51.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 52: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/52.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?
y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 53: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/53.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 54: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/54.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?
y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 55: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/55.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 56: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/56.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx
= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 57: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/57.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 58: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/58.jpg)
Example 4
Example
Find y′ if y = sin4 2xWhat is the inside function?u = sin 2xWhat is the derivative?u′ = cos 2x · d
dx 2xu′ = 2 cos 2x
What is the outside function?y = u4
What is the derivative?y′ = 4u3
Putting this together, we have
dydx
=dydu· du
dx= 4u3 · (2 cos 2x)
= 8 sin3 2x · cos 2x
![Page 59: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/59.jpg)
Exponential Functions
What do exponential functions look like when we graph them?
What do the equations of exponential functions look like?
f (x) = cax
We will focus here on functions of the form f (x) = ax for a ∈ R,a > 0.
![Page 60: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/60.jpg)
Exponential Functions
What do exponential functions look like when we graph them?
What do the equations of exponential functions look like?
f (x) = cax
We will focus here on functions of the form f (x) = ax for a ∈ R,a > 0.
![Page 61: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/61.jpg)
Exponential Functions
What do exponential functions look like when we graph them?
What do the equations of exponential functions look like?
f (x) = cax
We will focus here on functions of the form f (x) = ax for a ∈ R,a > 0.
![Page 62: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/62.jpg)
Exponential Functions
What do exponential functions look like when we graph them?
What do the equations of exponential functions look like?
f (x) = cax
We will focus here on functions of the form f (x) = ax for a ∈ R,a > 0.
![Page 63: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/63.jpg)
Exponential Functions
We want to find the derivative of f (x) = ax. What is the onlytechnique we know at this point?
ddx
ax = limh→0
ax+h − ax
h
= limh→0
ax(ah − 1)h
= ax limh→0
ah − 1h
Now what?
Let’s look at this numerically.
![Page 64: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/64.jpg)
Exponential Functions
We want to find the derivative of f (x) = ax. What is the onlytechnique we know at this point?
ddx
ax = limh→0
ax+h − ax
h
= limh→0
ax(ah − 1)h
= ax limh→0
ah − 1h
Now what?
Let’s look at this numerically.
![Page 65: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/65.jpg)
Exponential Functions
We want to find the derivative of f (x) = ax. What is the onlytechnique we know at this point?
ddx
ax = limh→0
ax+h − ax
h
= limh→0
ax(ah − 1)h
= ax limh→0
ah − 1h
Now what?
Let’s look at this numerically.
![Page 66: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/66.jpg)
Exponential Functions
We want to find the derivative of f (x) = ax. What is the onlytechnique we know at this point?
ddx
ax = limh→0
ax+h − ax
h
= limh→0
ax(ah − 1)h
= ax limh→0
ah − 1h
Now what?
Let’s look at this numerically.
![Page 67: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/67.jpg)
Exponential Functions
We want to find the derivative of f (x) = ax. What is the onlytechnique we know at this point?
ddx
ax = limh→0
ax+h − ax
h
= limh→0
ax(ah − 1)h
= ax limh→0
ah − 1h
Now what?
Let’s look at this numerically.
![Page 68: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/68.jpg)
Exponential Functions
We want to find the derivative of f (x) = ax. What is the onlytechnique we know at this point?
ddx
ax = limh→0
ax+h − ax
h
= limh→0
ax(ah − 1)h
= ax limh→0
ah − 1h
Now what?
Let’s look at this numerically.
![Page 69: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/69.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01 .695.001 .6934
Does this number look familiar to anyone?
![Page 70: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/70.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1
.7177.01 .695.001 .6934
Does this number look familiar to anyone?
![Page 71: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/71.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177
.01 .695.001 .6934
Does this number look familiar to anyone?
![Page 72: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/72.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01
.695.001 .6934
Does this number look familiar to anyone?
![Page 73: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/73.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01 .695
.001 .6934
Does this number look familiar to anyone?
![Page 74: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/74.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01 .695.001
.6934
Does this number look familiar to anyone?
![Page 75: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/75.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01 .695.001 .6934
Does this number look familiar to anyone?
![Page 76: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/76.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01 .695.001 .6934
Does this number look familiar to anyone?
![Page 77: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/77.jpg)
Derivative of f (x) = 2x
Let’s look at the limit part, since we know that the ax part is notdependent on the limit.
h 2h−1h
.1 .7177.01 .695.001 .6934
Does this number look familiar to anyone?
![Page 78: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/78.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01 1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 79: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/79.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1
1.746.01 1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 80: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/80.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746
.01 1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 81: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/81.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01
1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 82: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/82.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01 1.622
.001 1.611
What number are we approaching here? Is there a pattern?
![Page 83: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/83.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01 1.622.001
1.611
What number are we approaching here? Is there a pattern?
![Page 84: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/84.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01 1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 85: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/85.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01 1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 86: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/86.jpg)
Derivative of f (x) = 5x
Let’s try something similar for f (x) = 5x.
h 5h−1h
.1 1.746.01 1.622.001 1.611
What number are we approaching here? Is there a pattern?
![Page 87: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/87.jpg)
Derivative of f (x) = ax
.7177, .6955, .6934⇒
ln 2
1.746, 1.622, 1.611⇒ ln 5
and this makes our derivative ...
Derivative of f (x) = ax
ddx
ax = axln a
![Page 88: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/88.jpg)
Derivative of f (x) = ax
.7177, .6955, .6934⇒ ln 2
1.746, 1.622, 1.611⇒ ln 5
and this makes our derivative ...
Derivative of f (x) = ax
ddx
ax = axln a
![Page 89: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/89.jpg)
Derivative of f (x) = ax
.7177, .6955, .6934⇒ ln 2
1.746, 1.622, 1.611⇒
ln 5
and this makes our derivative ...
Derivative of f (x) = ax
ddx
ax = axln a
![Page 90: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/90.jpg)
Derivative of f (x) = ax
.7177, .6955, .6934⇒ ln 2
1.746, 1.622, 1.611⇒ ln 5
and this makes our derivative ...
Derivative of f (x) = ax
ddx
ax = axln a
![Page 91: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/91.jpg)
Derivative of f (x) = ax
.7177, .6955, .6934⇒ ln 2
1.746, 1.622, 1.611⇒ ln 5
and this makes our derivative ...
Derivative of f (x) = ax
ddx
ax = axln a
![Page 92: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/92.jpg)
Derivative of f (x) = ax
.7177, .6955, .6934⇒ ln 2
1.746, 1.622, 1.611⇒ ln 5
and this makes our derivative ...
Derivative of f (x) = ax
ddx
ax = axln a
![Page 93: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/93.jpg)
Hmmm ...
Is there an a such that ddx ax = ax?
Using what we just did, we want limh→0
ah−1h = 1.
Consider the following:
ah − 1h≈ 1
ah − 1 ≈ h
ah ≈ h + 1
a ≈ (1 + h)1h
![Page 94: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/94.jpg)
Hmmm ...
Is there an a such that ddx ax = ax?
Using what we just did, we want limh→0
ah−1h = 1.
Consider the following:
ah − 1h≈ 1
ah − 1 ≈ h
ah ≈ h + 1
a ≈ (1 + h)1h
![Page 95: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/95.jpg)
Hmmm ...
Is there an a such that ddx ax = ax?
Using what we just did, we want limh→0
ah−1h = 1.
Consider the following:
ah − 1h≈ 1
ah − 1 ≈ h
ah ≈ h + 1
a ≈ (1 + h)1h
![Page 96: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/96.jpg)
Hmmm ...
Is there an a such that ddx ax = ax?
Using what we just did, we want limh→0
ah−1h = 1.
Consider the following:
ah − 1h≈ 1
ah − 1 ≈ h
ah ≈ h + 1
a ≈ (1 + h)1h
![Page 97: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/97.jpg)
Hmmm ...
Is there an a such that ddx ax = ax?
Using what we just did, we want limh→0
ah−1h = 1.
Consider the following:
ah − 1h≈ 1
ah − 1 ≈ h
ah ≈ h + 1
a ≈ (1 + h)1h
![Page 98: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/98.jpg)
Hmmm ...
Is there an a such that ddx ax = ax?
Using what we just did, we want limh→0
ah−1h = 1.
Consider the following:
ah − 1h≈ 1
ah − 1 ≈ h
ah ≈ h + 1
a ≈ (1 + h)1h
![Page 99: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/99.jpg)
Hmmm ... (cont.)
Now let’s bring back the limit. Does this look familiar?
limh→0
(1 + h)1h
This may look more familiar in another form ...
limh→∞
(1 +
1h
)h
= e
Derivative of f (x) = ex
ddx
ex = ex
![Page 100: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/100.jpg)
Hmmm ... (cont.)
Now let’s bring back the limit. Does this look familiar?
limh→0
(1 + h)1h
This may look more familiar in another form ...
limh→∞
(1 +
1h
)h
=
e
Derivative of f (x) = ex
ddx
ex = ex
![Page 101: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/101.jpg)
Hmmm ... (cont.)
Now let’s bring back the limit. Does this look familiar?
limh→0
(1 + h)1h
This may look more familiar in another form ...
limh→∞
(1 +
1h
)h
= e
Derivative of f (x) = ex
ddx
ex = ex
![Page 102: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/102.jpg)
Hmmm ... (cont.)
Now let’s bring back the limit. Does this look familiar?
limh→0
(1 + h)1h
This may look more familiar in another form ...
limh→∞
(1 +
1h
)h
= e
Derivative of f (x) = ex
ddx
ex = ex
![Page 103: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/103.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 104: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/104.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?
u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 105: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/105.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3x
What is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 106: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/106.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?
u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 107: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/107.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 108: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/108.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?
y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 109: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/109.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 110: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/110.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?
y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 111: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/111.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 112: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/112.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx
= eu · 3= 3 e3x
![Page 113: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/113.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3
= 3 e3x
![Page 114: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/114.jpg)
Example 5
Example
Find f ′(x) if f (x) = e3x
What is the inside function?u = 3xWhat is the derivative?u′ = 3
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 3= 3 e3x
![Page 115: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/115.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 116: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/116.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?
u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 117: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/117.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1
What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 118: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/118.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?
u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 119: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/119.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 120: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/120.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?
y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 121: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/121.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
u
What is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 122: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/122.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?
y′ = 12√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 123: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/123.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 124: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/124.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 125: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/125.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 126: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/126.jpg)
Example 6
Example
Find f ′(x) for f (x) =√
ex + 1.
What is the inside function?u = ex + 1What is the derivative?u′ = ex
What is the outside function?y =√
uWhat is the derivative?y′ = 1
2√
u
Putting this together, we have
f ′(x) =dydu· du
dx
=1
2√
u· ex
=ex
2√
ex + 1
![Page 127: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/127.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 128: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/128.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?
u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 129: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/129.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 130: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/130.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?
u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 131: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/131.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 132: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/132.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?
y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 133: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/133.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 134: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/134.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?
y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 135: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/135.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 136: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/136.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx
= eu · 2x
= 2xex2
![Page 137: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/137.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 138: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/138.jpg)
Example 7
Example
Find f ′(x) for f (x) = ex2.
What is the inside function?u = x2
What is the derivative?u′ = 2x
What is the outside function?y = eu
What is the derivative?y′ = eu
Putting this together, we have
f ′(x) =dydu· du
dx= eu · 2x
= 2xex2
![Page 139: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/139.jpg)
Example 8
Example
Find where f (x) = 2x · e3x+1 has a horizontal tangent line.
What do we need to do here?
We need to use the product rule and the chain rule and set thederivative equal to 0.
![Page 140: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/140.jpg)
Example 8
Example
Find where f (x) = 2x · e3x+1 has a horizontal tangent line.
What do we need to do here?
We need to use the product rule and the chain rule and set thederivative equal to 0.
![Page 141: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/141.jpg)
Example 8
Example
Find where f (x) = 2x · e3x+1 has a horizontal tangent line.
What do we need to do here?
We need to use the product rule and the chain rule and set thederivative equal to 0.
![Page 142: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/142.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 143: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/143.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2
= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 144: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/144.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 145: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/145.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 146: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/146.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 147: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/147.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 148: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/148.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 149: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/149.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 150: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/150.jpg)
Example 8
f ′(x) = 2xddx
e3x+1 + e3x+1 ddx
2x
= 2x(3e3x+1)+ e3x+1 · 2= 6xe3x+1 + 2e3x+1
= e3x+1(6x + 2)
Can e3x+1 ever equal 0?
e3x+1(6x + 2) = 0
6x + 2 = 0
6x = −2
x = −13
![Page 151: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/151.jpg)
Example 9
Example
Find f ′(x) for f (x) =√
(x2 · 5x)3.
ddx
√(x2 · 5x)3 =
1
2√(x2 · 5x)3
· ddx
(x2 · 5x)3
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · ddx
(x2 · 5x)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 ·(
x2 ddx
5x + 5x ddx
x2)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · (x2 · 5x ln 5 + 5x · 2x)
![Page 152: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/152.jpg)
Example 9
Example
Find f ′(x) for f (x) =√
(x2 · 5x)3.
ddx
√(x2 · 5x)3
=1
2√(x2 · 5x)3
· ddx
(x2 · 5x)3
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · ddx
(x2 · 5x)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 ·(
x2 ddx
5x + 5x ddx
x2)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · (x2 · 5x ln 5 + 5x · 2x)
![Page 153: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/153.jpg)
Example 9
Example
Find f ′(x) for f (x) =√
(x2 · 5x)3.
ddx
√(x2 · 5x)3 =
1
2√
(x2 · 5x)3· d
dx(x2 · 5x)3
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · ddx
(x2 · 5x)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 ·(
x2 ddx
5x + 5x ddx
x2)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · (x2 · 5x ln 5 + 5x · 2x)
![Page 154: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/154.jpg)
Example 9
Example
Find f ′(x) for f (x) =√
(x2 · 5x)3.
ddx
√(x2 · 5x)3 =
1
2√
(x2 · 5x)3· d
dx(x2 · 5x)3
=1
2√
(x2 · 5x)3· 3(x2 · 5x)2 · d
dx(x2 · 5x)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 ·(
x2 ddx
5x + 5x ddx
x2)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · (x2 · 5x ln 5 + 5x · 2x)
![Page 155: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/155.jpg)
Example 9
Example
Find f ′(x) for f (x) =√
(x2 · 5x)3.
ddx
√(x2 · 5x)3 =
1
2√
(x2 · 5x)3· d
dx(x2 · 5x)3
=1
2√
(x2 · 5x)3· 3(x2 · 5x)2 · d
dx(x2 · 5x)
=1
2√
(x2 · 5x)3· 3(x2 · 5x)2 ·
(x2 d
dx5x + 5x d
dxx2)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · (x2 · 5x ln 5 + 5x · 2x)
![Page 156: 3.6 The Chain Rule - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/3.6_the_chain_rule.pdfWhat This Means When we have a composite function, we first identify the inside](https://reader035.vdocuments.mx/reader035/viewer/2022071401/60ea7a3383b8c51f867a5368/html5/thumbnails/156.jpg)
Example 9
Example
Find f ′(x) for f (x) =√
(x2 · 5x)3.
ddx
√(x2 · 5x)3 =
1
2√
(x2 · 5x)3· d
dx(x2 · 5x)3
=1
2√
(x2 · 5x)3· 3(x2 · 5x)2 · d
dx(x2 · 5x)
=1
2√
(x2 · 5x)3· 3(x2 · 5x)2 ·
(x2 d
dx5x + 5x d
dxx2)
=1
2√(x2 · 5x)3
· 3(x2 · 5x)2 · (x2 · 5x ln 5 + 5x · 2x)