Why does the product rule for differentiating functions work?

Let k(x) = f(x)g(x) then k'(x) = limit as h --> 0 of: (h(x+h)-h(x))/h = lim (f(x+h)g(x+h) - f(x)g(x))/h

We can factorise the numerator as:
f(x+h)g(x+h) - f(x)g(x) =f(x+h)g(x+h) -g(x)f(x+h) + g(x)f(x+h)- f(x)g(x) = f(x+h)[g(x+h) - g(x)] + g(x)[f(x+h)-f(x)]
Hence: 
lim (f(x+h)g(x+h) - f(x)g(x))/h = lim (f(x+h)[g(x+h) - g(x)] + g(x)[f(x+h)-f(x)]/h = lim (f(x+h)[g(x+h) - g(x)])/h + lim (g(x)[f(x+h)-f(x)])/h. We know that lim (f(x+h) - f(x) ) / h is just f'(x) by definition. And the same for g'(x). Hence we can replace these so:
k'(x) = lim h approaches 0 of f(x+h)g'(x) + g(x)f'(x). As h approaches 0, f(x+h) just approaches f(x) therefore we have k'(x) = f(x)g'(x) +g(x)h'(x)

This is all more easily explained on a board where we can keep clarity, but drop the brackets