if $g$ be a continuous function not differentiable at $0$ Let $g$ be a continuous function not differentiable at $0$ with $g(0)=8$. Let $f(x)=x\,g(x)$ .Find $f'(0)$  
a)$0$
b)$4$
c)$2$
d)$8$   
I am getting that $f'(x)=g(x)+x\,g'(x)$. But since $g'(x)$ doesn't exist for $x=0$, hence $f'(x)=8$. Please help whether it is right or wrong

It's wrong. You should try from the definition of differentiability.
Hint: $f^{'}(0)=\lim_{h \to 0} \frac{f(0+h)-f(0)}{h}$.
