How do I prove this using the definition of a derivative? If $f$ is a differentiable function and $g(x)=xf(x)$, use the definition of a derivative to show that $g'(x)=xf'(x)+f(x)$.

By definition $$ g'(x) = \lim_{\Delta x \to 0} \frac{g(x + \Delta x) - g(x)}{\Delta x} = \lim_{\Delta x \to 0} \frac{(x + \Delta x)f(x + \Delta x) - xf(x)}{\Delta x} = \lim_{\Delta x \to 0} \frac{x[f(x + \Delta x) - f(x)] + \Delta xf(x + \Delta x)}{\Delta x} = xf'(x) + f(x)$$
