Here we compute derivatives of products and quotients of functions

The product rule

Consider the product of two simple functions, say where and . An obvious guess for the derivative of is the product of the derivatives:

Is this guess correct? We can check by rewriting and and doing the calculation in a way that is known to work. Write with me

Hence so we see that So the derivative of is not as simple as . Never fear, we have a rule for exactly this situation.

Let’s return to the example with which we started.

Now that we are pros, let’s try one more example.

The quotient rule

We’d like to have a formula to compute This brings us to our next derivative rule.

It is often possible to calculate derivatives in more than one way, as we have already seen. Since every quotient can be written as a product, it is always possible to use the product rule to compute the derivative, though it is not always simpler.

Let’s finish up with one more, mixing the rules together.