15 B The Chain Rule

We now have a small list of “shortcuts” to find derivatives of simple functions. Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.

How would you go about finding the derivative of the following?

If h(x) = g(f(x)), then h’(x) = g’(f(x))●f’(x). The Chain Rule deals with the idea of composite functions and it is helpful to think about an outside and an inside function when using The Chain Rule.

Inside Function

Outside Function

In other words: The derivative when using the Chain Rule is the derivative of the outside leaving the inside unchanged times the derivative of the inside. If h(x) = g(f(x)), then h’(x) = g’(f(x))●f’(x).

Consider a simple composite function:

Find the derivative of Identify outside function and the inside function. The outside function is the cube, ( ) 3 The inside function is x The derivative of the inside using the Power Rule The derivative of the outside leaving the inside unchanged

Next, simplify

Find the derivative

Solutions

Find the derivative of To find the derivative of the outside, do the Power Rule:

Find the derivative of To find the derivative of the Inside, do the Power Rule:

Now do a little simplification: Multiply the 1/3 and the 6x. Now let’s look at the actual derivative using the Chain Rule. The derivative of the outside leaving the inside unchanged The derivative of the inside

One Last Thought It takes a big man to cry, but it takes a bigger man to laugh at that man.

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