## Differentiating Products of Functions

Many functions are built by multiplying two simpler functions. For example, is the product of the linear function and the function

On this page, we will verify the

Product Rule

As always, we'll compute the average rate of change of the function over the interval to This average rate of change is

Our argument will use the tangent line approximation to compute the approximate average rate of change. Remember that

and that these approximations are very good when is small.

If we multiply these two approximations together, we see that

This means that the average rate of change is

Now as becomes very small, the approximation improves and the last term above becomes increasingly small. This says that

Example:
Consider the function We may compute the derivative

Notice that in this example, the derivative depends on the value of in which we are interested.

Below, the graph of the function is shown on the left while its derivative is shown on the right.

Questions:
 Can you explain the relationship between these two graphs? What does it mean when the derivative is negative and what does it mean when it is positive? What happens at the point where the derivative equals zero?