03 - Chain Rule

Kuta Software - Infinite Calculus

Name___________________________________

Differentiation - Chain Rule

Date________________ Period____

Differentiate each function with respect to x.

1) y = ( x 3 + 3)

5

3) y = (?5 x 3 ? 3)

2) y = (?3 x 5 + 1)

3

4) y = (5 x 2 + 3)

4

5) f ( x) =

4

?3 x 4 ? 2

6) f ( x) =

7) f ( x) =

3

?2 x 4 + 5

8) y = (? x 4 ? 3)

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-1-

3

?2 x 2 + 1

?2

Worksheet by Kuta Software LLC

9) y = (3 x 3 + 1)(?4 x 2 ? 3) 4

10) y =

( x 3 + 4) 5

3x4 ? 2

11) y = (( x + 5) ? 1)

5

12) y = (5 x 3 ? 3) 5

4

4

?4 x 5 ? 3

Critical thinking question:

13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the

function.

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Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus

Name___________________________________

Differentiation - Chain Rule

Date________________ Period____

Differentiate each function with respect to x.

5

1) y = ( x 3 + 3)

dy

4

= 5( x 3 + 3) ? 3 x 2

dx

4

= 15 x 2 ( x 3 + 3)

3) y = (?5 x 3 ? 3)

2) y = (?3 x 5 + 1)

dy

2

= 3(?3 x 5 + 1) ? ?15 x 4

dx

2

= ?45 x 4 (?3 x 5 + 1)

3

4) y = (5 x 2 + 3)

dy

= 3(?5 x 3 ? 3) 2 ? ?15 x 2

dx

2

= ?45 x 2 (?5 x 3 ? 3)

5) f ( x) =

4

3

7) f ( x) =

3

? 2)

?

1

f ' ( x) = (?2 x 2 + 1) 2 ? ?4 x

2

2x

=?

3

4

(?2 x

?2 x 4 + 5

8) y = (? x 4 ? 3)

2

3(?2 x + 5)

2

3

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2

+ 1)

1

2

?2

dy

= ?2(? x 4 ? 3) ?3 ? ?4 x 3

dx

8x3

=

(? x 4 ? 3) 3

?

1

f ' ( x) = (?2 x 4 + 5) 3 ? ?8 x 3

3

8x3

=?

4

?2 x 2 + 1

1

?

1

f ' ( x) = (?3 x 4 ? 2) 4 ? ?12 x 3

4

3x3

=?

(?3 x

4

dy

= 4(5 x 2 + 3) 3 ? 10 x

dx

3

= 40 x(5 x 2 + 3)

6) f ( x) =

?3 x 4 ? 2

4

3

-1-

Worksheet by Kuta Software LLC

9) y = (3 x 3 + 1)(?4 x 2 ? 3) 4

dy

3

4

= (3 x 3 + 1) ? 4(?4 x 2 ? 3) ? ?8 x + (?4 x 2 ? 3) ? 9 x 2

dx

3

= x(?4 x 2 ? 3) (?132 x 3 ? 32 ? 27 x)

10) y =

( x 3 + 4) 5

3x4 ? 2

4

5

dy (3 x 4 ? 2) ? 5( x 3 + 4) ? 3 x 2 ? ( x 3 + 4) ? 12 x 3

=

dx

(3 x 4 ? 2) 2

4

3 x 2 ( x 3 + 4) (11 x 4 ? 10 ? 16 x)

=

(3 x 4 ? 2) 2

11) y = (( x + 5) ? 1)

5

4

3

dy

5

4

= 4(( x + 5) ? 1) ? 5( x + 5)

dx

3

= 20(( x + 5) ? 1) ? ( x + 5)

5

12) y = (5 x 3 ? 3) 5

4

4

?4 x 5 ? 3

3

1

?

1

dy

= (5 x 3 ? 3) 5 ? (?4 x 5 ? 3) 4 ? ?20 x 4 + (?4 x 5 ? 3) 4 ? 5(5 x 3 ? 3) 4 ? 15 x 2

dx

4

2

3

5 x (5 x ? 3) 4 (?65 x 5 ? 45 + 3 x 2 )

=

(?4 x

5

? 3)

3

4

Critical thinking question:

13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the

function.

(

6

)

7

Many answers: Ex y = ((2 x + 1) + 2) + 3

6

6

5

dy

5

5

4

= 7 ((2 x + 1) + 2) + 3 ? 6((2 x + 1) + 2) ? 5(2 x + 1) ? 2

dx

(

5

)

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