7.7 The Chain Rule

A. What’s a Composition of FunctionsB. The Chain Rule

C. When a Problem Uses More than One Rule

A. What’s a composition of functions?

• You’ll have an “inside function” and an “outside function” sort of.

• We will call the outside function f and the inside function g.

• When you plug the function g INTO f, you get the original composition of functions.

• You will understand these words better if I show you….

( ) ( )( )( )

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4 :gInsert

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f. INTO g PLUG means This

. Find

.4 and

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2

x

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xxgxxf

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( )

( )

( )

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2

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=

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x

x

B. The Chain Rule( ) ( )

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12 −⋅−=−

′⋅′=

xxdx

d

ggfgfdx

d

( )( ) =+ 52 1xdx

d

( ) ( ). Find .175 xfxxxf ′+−=

( ) =−− 43 34 xxdx

d

3

2 1

1

+xdx

d

( )( )2/13 −− xxdx

d

( )

+3 242

1 :You try

xdx

d

3 4227 :You try +x

C. When a problem uses more than one rule

( ) ( )[ ] =+− 74 2925 xxdx

d

4

1

+xx

dx

d

( )[ ] 5 322 1−+ zzdz

d

( ) ( )[ ]43 1514 :You try +− xxdx

d

3

2

2

3 :You try

+x

x

dx

d

( )[ ]4323 1 :You try xxdx

d ++