if f and g are differentiable functions, then
d dx f(x) g(x) = g(x) f'(x)f(x)g'(x) (g(x))2
Using the limit definition of the derivatives, prove this derivative rule. Here are the basics steps:
• Use the limit definition of the derivative.
• Get a common denominator for the functions in the numerator.
• Add and subtract f(x)g(x) in the numerator. Why can we do this?
• Factor out 1 g(x+h)g(x)
• Split the numerator into two fractions one should contain g(x+h)g(x) f(x+h)-f(x) and the other should contain Note that both of these fractions contain another term that is multiplied to them.
