Let $f$ and $g$ be two differentiable functions.

1. Show that

   $$\begin{array}{rl}& \frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\ =& g(x+h)\left(\frac{f(x+h)-f(x)}{h}\right)+f(x)\left(\frac{g(x+h)-g(x)}{h}\right)\end{array}$$

2. Hence, prove the product rule from first principles.

For (a), it is easier to start from the right-hand side and prove this is equal to the left-hand side

By first principles, if

then