If f and g are differentiable at x, the derivative of f×g at x is f′(x)×g(x) + f(x)×g′(x). This is known as the product rule for differentiation.
Write the derivative of the product as the following limit, where fg is the function f times the function g.
fg(x+h) - fg(x) over h
Subtract a limit which is known to be 0, namely h times the product of the individual derivatives, or:
h × (f(x+h)-f(x))/h × (g(x+h)-g(x))/h
After h and h cancel, the common denominator is h throughout, hence we can concentrate on the numerators. The first term in the above product is fg(x+h), which neatly cancels fg(x+h) in the original expression for the derivative. The last term is fg(x), which is subtracted from -fg(x), giving -2fg(x). Bring in the mixed terms and regroup, giving the difference quotient that corresponds to f′(x)×g(x) + g′(x)×f(x).