7. Differentiating Powers of a Function

by M. Bourne

Function of a Function

If y is a function of u, and u is a function of x, then we say

"y is a function of the function u".

Example 1

Consider the function

y = (5x + 7)¹².

If we let u = 5x + 7 (the inner-most expression), then we could write our original function as

y = u¹²

We have written y as a function of u, and in turn, u is a function of x.

This is a vital concept in differentiation, since many of the functions we meet from now on will be functions of functions, and we need to recognise them in order to differentiate them properly.

Chain Rule

To find the derivative of a function of a function, we need to use the Chain Rule:

`(dy)/(dx) = (dy)/(du) (du)/(dx)`

This means we need to

1. Recognise `u` (always choose the inner-most expression, usually the part inside brackets, or under the square root sign).
2. Then we need to re-express `y` in terms of `u`.
3. Then we differentiate `y` (with respect to `u`), then we re-express everything in terms of `x`.
4. The next step is to find `(du)/dx`.
5. Then we multiply `dy/(du)` and `(du)/dx`.

Answer for Example 1

In Example 1,

y = u¹², so `(dy)/(du) = 12u^11 = 12(5x+7)^11`

and

u = 5x + 7, so `(du)/(dx) = 5`

So

`dy/dx=(dy)/(du) (du)/(dx) ` `= (12(5x+7)^11)(5) ` `= 60(5x+7)^11`

Example 2

Find `dy/dx` if `y = (x^2+ 3)^5`.

Show answer

Play with the graph of this example on the Differentiation interactive applet page and explore what it means.

Example 3

Find `dy/dx` if `y=sqrt(4x^2-x)`.

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You can play with this example on the Differentiation interactive applet page.

The Derivative of a Power of a Function (Power Rule)

An extension of the chain rule is the Power Rule for differentiating. We are finding the derivative of uⁿ (a power of a function):

`d/dxu^n=n u^(n-1)(du)/dx`

Example 4

In the case of `y=(2x^3-1)^4` we have a power of a function.

Show answer

Play with this example on the Differentiation interactive applet page.

Example 5

If `y=1/x`, find `dy/dx`

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You can explore this example on the Differentiation interactive applet page.

CHALLENGE

Find the derivative of `y=(x^2(3x+1))/(x^4+2)`

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Play with this challenge example on the Differentiation interactive applet page.