I am having trouble with this problem. It asks to: Prove that the function f(x)= x^2*sin(1/x) if x doesn't equal zero and 0 if x=0 is differentiable at x=0 and to compute the derivative at zero.

This is what i've done so far:

Noted f(0)=0 so its defined at 0.

Proved that the limit of f(x) as x->0 is 0 through the squeeze theorem.

-Through this f(x) is continuous*

I think i need to prove that the limit of the derivative from the left and right as x->0 = some number correct? or am i wrong? this way it would prove no sharp turns and the derivative at 0? Can i get some hints or help? Thank you!