IXL - Find derivatives using the chain rule II (Calculus practice)

Find the derivative of

(

)

(

)

(

)

′

(

)

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Learn with an example

question

Find the derivative of

(

)

(

)

[

–

cos

(

)

]

′

(

)

key idea

The chain rule says that if

(

)

and

(

)

are differentiable, then their composition

(

)

can be differentiated by the following formula:

[

(

)

]

′

(

)

′

(

)

Common derivatives

u ( x )	u ′ ( x )
x n	n x n – 1
sin ( x )	cos ( x )
cos ( x )	– sin ( x )
tan ( x )	[ sec ( x ) ] 2
csc ( x )	– csc ( x ) cot ( x )
sec ( x )	sec ( x ) tan ( x )
cot ( x )	– [ csc ( x ) ] 2

solution

The function

(

)

[

–

cos

(

)

]

is a composition of two functions:

(

)

and

(

)

–

cos

(

)

The outer function is

(

)

and the inner function is

(

)

–

cos

(

)

(

)

[

(

)

]

[

–

cos

(

)

]

To find

′

(

)

apply the chain rule. The chain rule says to take the derivative of the outer function evaluated at the inner function, then multiply by the derivative of the inner function.

The outer function

(

)

has derivative

′

(

)

This derivative evaluated at the inner function is

′

(

)

[

–

cos

(

)

]

So:

′

(

)

(

[

–

cos

(

)

]

)

′

(

[

–

cos

(

)

]

)

[

–

cos

(

)

]

′

To complete this calculation, find the derivative of the inner function

(

)

–

cos

(

)

′

(

)

[

–

cos

(

)

]

′

(

)

′

–

[

cos

(

)

]

′

(

)

–

[

–

sin

(

)

]

sin

(

)

Now, put it all together.

′

(

)

(

[

–

cos

(

)

]

)

[

–

cos

(

)

]

′

(

[

–

cos

(

)

]

)

[

sin

(

)

]

So,

′

(

)

(

[

–

cos

(

)

]

)

[

sin

(

)

]

Looking for more? Try this:

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