Given that f(x) = (x^2 + 3)(5 - x), find f'(x).

First we must multiply out the brackets, using FOIL (first, outer, inner, last). This gives f(x) = -x^3 + 5x^2 - 3x + 15. Alternatively you could leave the brackets as they are, and use the product rule to differentiate the expression as it is.

The next step is to differentiate the function. To achieve this, we first multiply each term in the expression by its power of x, then reduce its power of x by 1. For the 15 term, the power of x is 0, so multiplying by 0 removes the term entirely.

If using the product rule d/dx(ab) = adb/dx + b*da/dx, we must differentiate each bracket and multiply by the other bracket. Afterwards we must combine the two terms, and group similar powers of x.

After this step, we finish with f'(x) = -3x^2 + 10x - 3.