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There's a general product rule you might not have heard about which can simplify some things:
whenever you have a function f(x) in which there are more than 1 x's, divide the group of x's in 2 groups. Differentiate keeping one group of x's constant, and add to that the derivative keeping the other group of x's constant. The sum is the derivative.
For example:
to calculate the derivative of x^x, you keep the exponent x constant to get x (x^(x-1))=x^x, then the base constant to get ln(x)x^x, add them up to get (x^x)' = x^x (ln(x) + 1). Easy.
Another example: group 2 x's in the following expression: f(x)= x^3 = x* x^2. Then f'(x) = x^2 + x * (2x) = 3x^2. This works literally every time.
