MAT 1221Survey of Calculus

Section 2.5

The Chain Rule

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Expectations

Use equals signs Pay attention to units Problem 2: Formally answer the question

Reminder

Exam 1, next Monday• Look at the comments from the grader• Look at the solutions online• Make improvements

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

Recall (Composite Functions)

Suppose2( ) , ( ) 1u u gf x xu

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

The Chain Rule: Part I

If and are both differentiable, then

is also differentiable

gfF

The Chain Rule: Part II Version 1

Meaning Find and then substitute Multiply by

)())(()( xgxgfxF

)(uf )(xgu

)(xg

( ) ( ( ))F x f g x

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

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

Special Case:Extended Power Rule

dx

dunu

dx

dy

xguuy

xgy

n

n

n

1

)( ,

)(

u

Special Case:Extended Power Rule

dx

dunu

dx

dy

xguuy

xgy

n

n

n

1

)( ,

)(

u

Special Case:Extended Power Rule

( )

, ( )

n

n

y g x

y u u g x

dy dy du

dx du dx

u

1

3 3 21 1y x x

Example 1

1

3 3 21 1y x x

Example 1 u

1 n ndy duy u nu

dx dx

Expectations

Must show the “formula” step when using the Chain Rule/ Extended Power Rule

1

3 321

( ) 1 12

f x x x

2 3( 1)y x x

Example 2

Example 2

2 3( 1)y x x

u

1 n ndy duy u nu

dx dx

Example 3 (With Product Rule)

2 1 3 1y x x

1 n ndy duy u nu

dx dx

Example 4 (With Quotient Rule)

3 1

2 1

xy

x

1 n ndy duy u nu

dx dx