Given the piecewise function

g(x) =

{ x^(3)e^(-x^(2)/4)sin(4/x^(2)) if x=/=0

{ 0 if x = 0

Prove that g is differentiable and g' is bounded on (-infinity,infinity).

I know that I have to use the definition of a derivative for the first part (not too sure about how to do that). How to show it's bounded, I am not sure.

This is the first time I have ever seen anything like this.

Thanks for any help.