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MAT 1221Survey of Calculus
Section 2.5
The Chain Rule
http://myhome.spu.edu/lauw
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Expectations
Use equals signs Pay attention to units Problem 2: Formally answer the question
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Reminder
Exam 1, next Monday• Look at the comments from the grader• Look at the solutions online• Make improvements
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Preview
We know how to find derivatives of Power functions, polynomials, products, quotients.
Section 2.5: Composite functions We will look at the chain rule. Focus on the special case : Extended
Power Rule
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Recall (Composite Functions)
Suppose2( ) , ( ) 1u u gf x xu
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Recall (Composite Functions)
Suppose Then,
2( ) , ( ) 1u u gf x xu
2
2
( )( ) ( )
( )
( ) 1
(
1
) ?
f g x f
f
F
x
x
x
x
x
g
F
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The Chain Rule: Part I
If and are both differentiable, then
is also differentiable
gfF
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The Chain Rule: Part II Version 1
Meaning Find and then substitute Multiply by
)())(()( xgxgfxF
)(uf )(xgu
)(xg
( ) ( ( ))F x f g x
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The Chain Rule: Part II Version 2
( ), ( )
Therefore, ( )( )
y f u u g x
y f g x
dy dy du
dx du dx
xdx
du
udx
dydu
dyy
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The Chain Rule: Part II Version 2
xdx
du
udx
dydu
dyy
( ), ( )
Therefore, ( )( )
y f u u g x
y f g x
dy dy du
dx du dx
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Special Case:Extended Power Rule
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
u
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Special Case:Extended Power Rule
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
u
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Special Case:Extended Power Rule
( )
, ( )
n
n
y g x
y u u g x
dy dy du
dx du dx
u
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1
3 3 21 1y x x
Example 1
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1
3 3 21 1y x x
Example 1 u
1 n ndy duy u nu
dx dx
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Expectations
Must show the “formula” step when using the Chain Rule/ Extended Power Rule
1
3 321
( ) 1 12
f x x x
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2 3( 1)y x x
Example 2
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Example 2
2 3( 1)y x x
u
1 n ndy duy u nu
dx dx
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Example 3 (With Product Rule)
2 1 3 1y x x
1 n ndy duy u nu
dx dx
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Example 4 (With Quotient Rule)
3 1
2 1
xy
x
1 n ndy duy u nu
dx dx