2 The derivative of a function of a function

To differentiate a function of a function we use the following Key Point:

Key Point 11

The Chain Rule

If y = f ( g ( x ) ) , that is, a function of a function, then

d y d x = d f d g × d g d x
This is called the chain rule .
Example 12

Find the derivatives of the following composite functions using the chain rule and check the result using other methods

  1. ( 2 x 2 − 1 ) 2
  2. ln e x
Solution
  1. Here y = f ( g ( x ) ) where f ( g ) = g 2 and g ( x ) = 2 x 2 − 1 . Thus

    d f d g = 2 g and d g d x = 4 x ∴ d y d x = 2 g . ( 4 x ) = 2 ( 2 x 2 − 1 ) ( 4 x ) = 8 x ( 2 x 2 − 1 )

    This result is easily checked by using the rule for differentiating products:

    y = ( 2 x 2 − 1 ) ( 2 x 2 − 1 ) so d y d x = 4 x ( 2 x 2 − 1 ) + ( 2 x 2 − 1 ) ( 4 x ) = 8 x ( 2 x 2 − 1 ) as obtained above.

  2. Here y = f ( g ( x ) ) where f ( g ) = ln g and g ( x ) = e x . Thus

    d f d g = 1 g and d g d x = e x ∴ d y d x = 1 g ⋅ e x = 1 e x ⋅ e x = 1

    This is easily checked since, of course,

    y = ln e x = x   and so, obviously  d y d x = 1  as obtained above.

Task!

Obtain the derivatives of the following functions

  1. ( 2 x 2 − 5 x + 3 ) 9
  2. sin ( cos x )
  3. 2 x + 1 2 x − 1 3
  1. Specify f and g for the first function:

    f ( g ) = g 9 g ( x ) = 2 x 2 − 5 x + 3 Now obtain the derivative using the chain rule:

    9 ( 2 x 2 − 5 x + 3 ) 8 ( 4 x − 5 ) . Can you see how to obtain the derivative without going through the intermediate stage of specifying f , g ?

  2. Specify f and g for the second function:

    f ( g ) = sin g g ( x ) = cos x Now use the chain rule to obtain the derivative:

    − [ cos ( cos x ) ] sin x

  3. Apply the chain rule to the third function:

    − 12 ( 2 x + 1 ) 2 ( 2 x − 1 ) 4