These are some simple examples of applications of the product rule and the quotient rule. In the first problem there is another way of finding the derivative: distributing the product. But using the product rule is faster.

First, using the product rule, we put:

So,

Expanding this product we get (this last step is not required, I just do it to check the answer later):

Now, using the direct approach, that is, exanding the product in the original function:

And taking the derivative of that:

And the two answers are the same.

Next there is a similar application of the quotient rule. However, as I said in the quotient rule page, I prefer to just use the product rule instead of memorizing another formula. We have:

We can write this as:

And now we apply the product rule plus the chain rule:

So we have: