How do I use product rule when differentiating?

Problems like this usually look something like: Find the differential of (x² e^2x). The product rule is used when we have to differentiate two different functions multiplied by eachother, so we should always look for this when doing questions. In this example, the functions are f(x) = x² , and g(x) = e^2x. To differentiate this, we have to find f'(x) and g'(x), that is, the differential of the two functions. The product rule is f'(x)*g(x)+g'(x)*f(x). This looks daunting but it's not as bad as it seems.
The easy way to look at these questions is to just split the thing you're given into two parts, then differentiate those parts, and apply the formula. Like all maths questions the only way to get good is to practice a lot. So heres some examples!...