Product rule for derivatives

Write $\frac{f(x+h)g(x+h)-f(x)g(x)}{h}$ and insert $-f(x+h)g(x)+f(x+h)g(x)$ in the numerator to split it as $f(x+h)\cdot\frac{g(x+h)-g(x)}{h} + \frac{f(x+h)-f(x)}{h}\cdot g(x)$. As $h\to 0$, the first difference quotient tends to $g'(x)$ and $f(x+h)\to f(x)$ by continuity (since $f$ is differentiable at $x$), while the second difference quotient tends to $f'(x)$; the sum tends to $f(x)g'(x)+f'(x)g(x)$.