Related to the Product Rule is the Quotient Rule.

$$ f'(  {f(x)  \over g(x)}  ) =  {{ g(x)   f'( f(x) )  -  f(x)   f'( g(x) ) } } \over {g(x)^2} $$

The derivative of the quotientof two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all over the denominator squared.