Derivative using limit definition Given that f is a differentiable function and g(x) = xf(x). Use the limit definition of derivative show that g'(x) = xf'(x) + f(x).
I understand that you have to find the derivative of xf(x) using the difference quotient 
but when I set up the problem I can't really simplify it.
Any ideas?
This is what I have so far:
Lim as h -> 0
(x+h) f(x+h) - xf(x) / h

Let $g(x)=xf(x)$. we want to calculate
$$\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}.$$
Substituting, we find that we want 
$$\lim_{h\to 0} \frac{(x+h)f(x+h)-xf(x)}{h}.$$
Expand. We want
$$\lim_{h\to 0} \left(\frac{xf(x+h)-xf(x)}{h}+f(x+h)\right).$$
This is
$$x\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}+\lim_{h\to 0} f(x+h).$$
To finish, note that since $f$ is differentiable at $x$, it is continuous at $x$, so $\lim_{h\to 0}f(x+h)=f(x)$.  
