How do I use the product rule for differentiation?

You should use the product rule when you have a function f(x), which you can't differentiate straight away. But which can be written in the form f(x)=g(x)h(x), where g(x) and h(x) are functions that you do know how to differentiate. Then f'(x)= g(x)h'(x)+h(x)g'(x). This may seem very abstract but an example will make it clear how to use this in practice. Say we wanted to differentiate f(x)=xe^x. At first glance this appears difficult. It is not a 'standard' function which we know how to differentiate. But we see if we set g(x)=x and h(x)=e^x, that f is of the required form to apply the product rule. Because we know that g'(x)=1, and h'(x)=e^x. So applying the product rule we see f'(x)=(1)(e^x)+x(e^x). Which we can simplify to e^x(1+x).