Chapter Contents

• Differentiation
• 1. Limits and Differentiation
• 2. The Slope of a Tangent to a Curve (Numerical)
• 3. The Derivative from First Principles
• 4. Derivative as an Instantaneous Rate of Change
• 5. Derivatives of Polynomials
• 6. Derivatives of Products and Quotients
• Derivative interactive graph
• Differentiation Java applet
• Derivative of a Cubic - Exploration using GeoGebra
• 7. Differentiating Powers of a Function
• 8. Differentiation of Implicit Functions
• 9. Higher Derivatives
• 10. Partial Derivatives
• Differentiation Problem Solver

• Comments, Questions?

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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`.

Example 2

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

Answer

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

Example 3

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

Answer

You can play with this example on the Differentiation Java 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.

Answer

Play with this example on the Differentiation Java applet page.

CHALLENGE

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

Answer

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