How would I differentiate a function such as f(x)=x^3(e^(2x))?

Here, f(x)=x³e^2x is a function consisting of two functions multiplied together, so we need to use the product rule. The product rule is as follows: where f(x)=u(x)v(x), f'(x)=u(x)v'(x)+u'(x)v(x). The first step involves identifying the two functions that are multiplied together, and representing them by u and v. So, let u(x)=x³ and v(x)=e^2x. Now, we must find u'(x) and v'(x). u'(x)=3x² (from Core 1: multiply by the power, then subtract 1 from the power) and v'(x)=2e^2x  (from using the chain rule). Then, substitute u(x), v(x), u'(x) and v'(x) into our product rule formula, giving f'(x)=x³ . 2e^2x + 3x² .e^2x     If you wish to simplify this, you can do so by taking out a common factor of x²e^2x from each term: f'(x)=x²e^2x(2x+3)