In order to use the product rule, we need to write our function as a product of two other functions. There are several ways to do this. We can write f(x) = (x^{2}e^{x})(sin x) or we can write f(x) = (x^{2})(e^{x}sin x). Use the first version, f(x) = (x^{2}e^{x})(sin x). In order to use the product rule we need to find the derivative of each factor. (sin x)' = cos x. For the other, we need to use the product rule again: Now we need to remember to put everything back together to find the derivative of the original function: We have several choices of what to do with the final answer. The answer f ' (x) = (2xe^{x} + x^{2}e^{x})sin x + x^{2}e^{x}cos x is a reasonable way to write it, since we can look at this answer and see that it came from the product rule. Another reasonable thing to do is factor out common factors to find f ' (x) = xe^{x}[(2 + x)sin x + xcos x], which might be helpful if we wanted to do something with the derivative later. We could also multiply everything out to find f ' (x) = 2xe^{x}sin x + x^{2}e^{x}sin x + x^{2}e^{x}cos x. The product rule also allows us to find derivatives of more complicated products. To find the derivative of 3 or more functions multiplied together, we need to be careful with our work, so that we don't lose track of things. |