let g(x) be such that l g(x) l <= M for all x in [-1,1]
let h(x) = (x^2)g(x) if x is not equal to 0
= 0 if x is equal to 0
show that h(x) is diffrentiable at x = 0 and find h'(0)
Remember that $\displaystyle h'(0)= \lim_{x\rightarrow 0} \frac{h(x)-h(0)}{x-0} =\lim_{x\rightarrow 0} \frac{h(x)}{x}$ and to evaluate the limit we use the squeeze theorem.