Differentiate y = xe^(2x).

We want to find dy/dx. We find this using the product rule by setting the functions f(x) = x and g(x) = e^2x. With these functions, we can write the equation as y = f(x)g(x), so by applying the product rule, we have that dy/dx = f'(x)g(x) + f(x)g'(x). To calculate g'(x), we use the chain rule.If we write h(x) = 2x, then g(x) = e^2x = e^h(x). So by using the chain rule and the fact that e^x differentiates to itself, we have that g'(x) = h'(x)e^h(x) = 2e^2x. Therefore by going back to the equation which we found by the product rule, dy/dx = f'(x)g(x) + f(x)g'(x) = (1)(e^2x) + (x)(2e^2x) = e^2x + 2xe^2x. We can factorise this to get dy/dx = (1 + 2x)e^2x.