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
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In this section:
Chain Rule
Power Rule
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:
This means we need to
- Recognise u (always choose the inner-most expression, usually the part inside brackets, or under the square root sign).
- Then we need to re-express y in terms of u.
- Then we differentiate y (with respect to u), then we re-express everything in terms of x.
- The next step is to find du/dx.
- Then we multiply dy/du and du/dx.
Example 1:
Find 
if y =
(x2+
3)5.
Answer
In this case, we let u = x2 + 3 and then y = u5.
We see that:
- u is a function of x and
- y is a function of u.
For the chain rule, we firstly need to find 
and 
.
So
Play with the graph of this example on the Differentiation Java applet page and explore what it means.
Example 2:
Find if 
.
In LiveMath, we get the following result. You can change the function to see what happens, or to check your own work.
Now for the normal 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):
Example:
In the case of 
we have a
power of a function.
In LiveMath, we get the following result.
Play with this example on the Differentiation Java applet page.
CHALLENGE
Find the derivative of
In LiveMath, we get the following result. Once again, you can check your own calculations by changing the function.
Play with this challenge example on the Differentiation Java applet page.
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