prove, from the definition of the derivative that the derivative of the product of two functions (fg)' equal f'g + g'f
Assume that f and g are both differentiable at some point c.
$\displaystyle (fg)'(c) = \lim_{x \to c} \frac{(fg)(x) - (fg)(c)}{x-c} = \lim_{x \to c} \frac{f(x)g(x) - f(c)g(c)}{x-c} = \lim_{x \to c} f(x) \frac{g(x) - g(c)}{x-c} $ $\displaystyle + \lim_{x \to c} g(c) \frac{f(x) - f(c)}{x-c} $
and since a function is continuous where it is differentiable
$\displaystyle = f(c)g'(c) + g(c)f'(c) $