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C CONVERSATION: Voice level 0. No talking!
HHELP: Raise your hand and wait to be called on.
AACTIVITY: Whole class instruction; students in seats.
M MOVEMENT: Remain in seat during instruction.
PPARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed.NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK.
S
Activity: Teacher-Directed Instruction
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Calculus AB
2013 Implicit Differentiation
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Objective
• C: The swbat differentiate implicitly equations in more than one variable.
• L: the sw explain to others how to find derivatives of multiple types of problems verbally and demonstratively
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Implicit Differentiation
Equation for a line:
Explicit Form
<One variable given explicitly in terms of the other>
Implicit Form
<Function implied by the equation>
Differentiate the Explicit
< Explicit: , y is function of x >
Differentiation taking place with respect to x. The derivative is explicit also.
y mx b
Ax By C
24 3 4y x x
8 3dy
xdx
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Implicit Differentiation
Equation of circle:
To work explicitly; must work two equations
2 2 9y x
29y x
Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly.
( Usually non-functions)
29y x 𝑑𝑦𝑑𝑥
=12
(9−𝑥2 )−12 (−2 𝑥 )
𝑑𝑦𝑑𝑥
=−𝑥
√9−𝑥2
𝑑𝑦𝑑𝑥
=− 12
( 9− 𝑥2 )− 12 (−2𝑥 )
𝑑𝑦𝑑𝑥
=𝑥
√9−𝑥2
𝑥2𝑦+2 𝑦2𝑥+3 𝑦3=7
Don’t want to solve for y
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Implicit Differentiation
Chain Rule Pretend y is some function like
so becomes
(A)
(B)
(C)
Note: Use the Leibniz form. Leads to Parametric and Related Rates.
2 2 3y x x 2 4( 2 3)x x 4y
Find the derivative with respect to x
< Assuming - y is a differentiable function of x >
32y
4y
2 3x y
=
=
2𝑑𝑥𝑑𝑥
+3𝑑𝑦𝑑𝑥
=2+3𝑑𝑦𝑑𝑥
4 𝑦 3(2𝑥+2)
6 𝑦2(2 𝑥+2)
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Implicit Differentiation
Find the derivative with respect to x
< Assuming - y is a differentiable function of x >
𝑥𝑦=¿ 𝑥𝑑𝑦𝑑𝑥
+𝑦𝑑𝑥𝑑𝑥 ¿ 𝑥
𝑑𝑦𝑑𝑥
+ 𝑦
𝑥2+ 𝑦2=¿2 𝑥𝑑𝑥𝑑𝑥
+2 𝑦𝑑𝑦𝑑𝑥
¿2 𝑥+2 𝑦𝑑𝑦𝑑𝑥
sin (𝑥𝑦) ¿cos (𝑥𝑦 )∗(𝑥 𝑑𝑦𝑑𝑥+𝑦 𝑑𝑥𝑑𝑥 )
¿ 𝑥cos (𝑥𝑦 ) 𝑑𝑦𝑑𝑥
+ 𝑦 cos (𝑥𝑦)
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Implicit Differentiation
(D) Product Rule
2xy ¿ 𝑥 (2 𝑦 ) 𝑑𝑦𝑑𝑥
+ 𝑦2 𝑑𝑥𝑑𝑥
¿2 𝑥𝑦𝑑𝑦𝑑𝑥
+𝑦2
¿ 𝑥𝑑𝑦𝑑𝑥
+ 𝑦❑ 𝑑𝑥𝑑𝑥
¿ 𝑥𝑑𝑦𝑑𝑥
+ 𝑦❑
𝑥𝑦
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Implicit Differentiation
(E) Chain Rule 3( )xy
Product inside a chain
3 (𝑥𝑦 )2(𝑥 𝑑𝑦𝑑𝑥
+𝑦𝑑𝑥𝑑𝑥
)
3 𝑥3 𝑦2 𝑑𝑦𝑑𝑥
+3 𝑥2 𝑦3
¿3 𝑥2 𝑦2(𝑥 𝑑𝑦𝑑𝑥+𝑦 𝑑𝑥𝑑𝑥 )
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sin (𝑥𝑦)
cos (𝑥𝑦 )∗(𝑥 𝑑𝑦𝑑𝑥 +𝑦 𝑑𝑥𝑑𝑥 )
𝑥cos (𝑥𝑦 ) 𝑑𝑦𝑑𝑥
+ 𝑦 cos(𝑥𝑦 )
Implicit Differentiation
(E) Chain Rule
Product inside a chain
𝑢=𝑥𝑦 𝑑𝑢=𝑥𝑑𝑦𝑑𝑥
+𝑦𝑑𝑥𝑑𝑥
sin𝑢 (𝑑𝑢)
cos𝑢(𝑥 𝑑𝑦𝑑𝑥+𝑦 𝑑𝑥𝑑𝑥 )
cos 𝑥𝑦 (𝑥 𝑑𝑦𝑑𝑥 )+cos 𝑥𝑦 (𝑦 𝑑𝑥𝑑𝑥 )
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Implicit Differentiation
To find implicitly.
EX: Diff Both Sides of equation with respect to x
Solve for
dy
dx
2 2 9x y dy
dx2 𝑥𝑑𝑥𝑑𝑥
+2 𝑦𝑑𝑦𝑑𝑥
=0
2 𝑥+2 𝑦𝑑𝑦𝑑𝑥
=0
𝑑𝑦𝑑𝑥
=−2𝑥2 𝑦
=−𝑥𝑦
29y x
29y x
Need both x and y to find the slope.
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C CONVERSATION: Voice level 0. No talking!
HHELP: Raise your hand and wait to be called on.
AACTIVITY: Whole class instruction; students in seats.
M MOVEMENT: Remain in seat during instruction.
PPARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed.NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK.
S
Activity: Teacher-Directed Instruction
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Objective
• C: The swbat differentiate implicitly equations in more than one variable.
• L: the sw explain to others how to find derivatives of multiple types of problems verbally and demonstratively
![Page 16: C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56649ea25503460f94ba5d68/html5/thumbnails/16.jpg)
EX 1:3 2 25 4y y y x
(a) Find the derivative at the point ( 5, 3 ) , at ( -1,-3 )
(b) Find where the curve has a horizontal tangent.
(c) Find where the curve has vertical tangents.
3 𝑦2 𝑑 𝑦𝑑𝑥
+2 𝑦𝑑𝑦𝑑𝑥−5
𝑑𝑦𝑑𝑥−2𝑥
𝑑𝑥𝑑𝑥
=0
(3 𝑦¿¿2+2 𝑦−5)𝑑𝑦𝑑𝑥
=2 𝑥¿
𝑑𝑦𝑑𝑥
=2𝑥
3 𝑦2+2 𝑦−5
𝑑𝑦𝑑𝑥 |¿ (5,3)=10
28𝑑𝑦𝑑𝑥 |¿ (−1 ,−3)=−2
16
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EX 1:3 2 25 4y y y x
(b) Find where the curve has a horizontal tangent.
Horizontal tangent has a 0 slope
𝑎𝑏
=0∴𝑎=0
2 𝑥=0
𝑥=0
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EX 1:3 2 25 4y y y x
(c) Find where the curve has vertical tangents. Vertical tangent has an undefined slope𝑎
𝑏𝑢𝑛𝑑𝑒𝑓 𝑏=0
3 𝑦2+2 𝑦−5=0
(3 𝑦+5)(𝑦−1)3 𝑦+5=0
𝑦=−53
𝑦−1=0
𝑦=1
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Ex 2:
3 3 2x y xy
< Folium of Descartes >
3 𝑥2 𝑑𝑥𝑑𝑥
+3 𝑦2 𝑑𝑦𝑑𝑥
=2(𝑥 𝑑𝑦𝑑𝑥+𝑦 𝑑𝑥𝑑𝑥 )
3 𝑥2+3 𝑦2 𝑑𝑦𝑑𝑥
=2𝑥𝑑𝑦𝑑𝑥
+2 𝑦
3 𝑦2 𝑑𝑦𝑑𝑥−2 𝑥
𝑑𝑦𝑑𝑥
=2 𝑦−3 𝑥2
𝑑𝑦𝑑𝑥 ( (3 𝑦
2−2 𝑥)(3 𝑦 2−2 𝑥))= 2 𝑦−3𝑥2
(3 𝑦2−2𝑥 )
𝑑𝑦𝑑𝑥
=2 𝑦−3𝑥2
3 𝑦2−2𝑥
3 𝑥2−2 𝑦=2 𝑥𝑑𝑦𝑑𝑥−3 𝑦2 𝑑𝑦
𝑑𝑥
3 𝑥2−2 𝑦=(2 𝑥−3 𝑦2)𝑑𝑦𝑑𝑥
3 𝑥2−2 𝑦(2 𝑥−3 𝑦2)
=𝑑𝑦𝑑𝑥
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Why Implicit?
3 3 2x y xy
< Folium of Descartes > Explicit Form:
3 6 3 3 6 33 31
1 1 1 18 8
2 4 2 4y x x x x x x
3 6 3 3 6 33 32 1
1 1 1 1 13 8 8
2 2 4 2 4y y x x x x x x
3 6 3 3 6 33 33 1
1 1 1 1 13 8 8
2 2 4 2 4y y x x x x x x
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2nd Derivatives
NOTICE:The second derivative is in terms of x , y , AND dy /dx.
The final step will be to substitute back the value of dy / dx into the second derivative.
EX: Our friendly circle. Find the 2nd Derivative.
2 2 9x y 2 𝑥
𝑑𝑥𝑑𝑥
+2 𝑦𝑑𝑦𝑑𝑥
=0
2 𝑥+2 𝑦𝑑𝑦𝑑𝑥
=0
2 𝑦𝑑𝑦𝑑𝑥
=−2 𝑥
𝑑𝑦𝑑𝑥
=−2𝑥2 𝑦
𝑑𝑦𝑑𝑥
=−𝑥𝑦
𝑑2 𝑦𝑑𝑥2 =¿
𝑦 (−1𝑑𝑥𝑑𝑥 )−(− 𝑥) 𝑑𝑦
𝑑𝑥
𝑦2
𝑑2 𝑦𝑑𝑥2 =
−𝑦+𝑥𝑑𝑦𝑑𝑥
𝑦 2
𝑑2 𝑦𝑑𝑥2 =
−𝑦+𝑥 (−𝑥𝑦 )𝑦2 ( 𝑦𝑦 )
−𝑦 2−𝑥2
𝑦3𝑑2 𝑦𝑑𝑥2 =¿
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2nd Derivatives
EX: Find the 2nd Derivative.
23 5xy
0−(𝑥 2 𝑦 𝑑𝑦𝑑𝑥
+𝑦2 𝑑𝑥𝑑𝑥 )=0
−2 𝑥𝑦𝑑𝑦𝑑𝑥− 𝑦2=0
−2𝑥 𝑑𝑦𝑑𝑥− 𝑦 (−2
𝑑𝑥𝑑𝑥 )
(−2𝑥)2
𝑑2 𝑦𝑑𝑥2 =¿
𝑑2 𝑦𝑑𝑥2 =¿
𝑑2 𝑦𝑑𝑥2 =¿
−2𝑥𝑑𝑦𝑑𝑥
+2 𝑦
4 𝑥2
−2𝑥 ( 𝑦−2𝑥 )+2 𝑦
4 𝑥2𝑑𝑦𝑑𝑥
= 𝑦 2
−2𝑥𝑦= 𝑦−2𝑥
𝑑2 𝑦𝑑𝑥2 =¿
𝑦+2 𝑦4 𝑥2
3 𝑦4 𝑥2
𝑑2 𝑦𝑑𝑥2 =¿
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Higher Derivatives
EX: Find the Third Derivative.
sin( )y x
cos (𝑦 ) 𝑑𝑦𝑑𝑥
=𝑑𝑥𝑑𝑥
𝑑𝑦𝑑𝑥
=1
cos(𝑦 )
𝑑𝑦𝑑𝑥
=sec (𝑦 )
𝑑2 𝑦𝑑𝑥2 =¿
𝑑2 𝑦𝑑𝑥2 =¿
𝑑2 𝑦𝑑𝑥2 =¿
sec (𝑦 ) tan (𝑦 )𝑑𝑦𝑑𝑥
sec (𝑦 ) tan ( 𝑦 ) sec (𝑦 )
𝑠𝑒𝑐2 (𝑦 ) tan (𝑦 )
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Last update
• 10/19/10
p. 162 1 – 29 odd