Calculate the derivative of $g(x)=x^2.cos(1/x^2)$. Does the derivative of this function exist at $x=0$? Calculate the derivative of $g(x)=x^2.cos(1/x^2)$.  Does the derivative of this function exist at $x=0$?
I calculated the derivative to be $2.cos(1/x^2)x+2(sin(1/x^2))/x$. 
I'm tempted to say that the derivative does not exist at zero because it is a vertical line, but I'm not sure this is the case.

$g'$ does not have limit in $0$, but it does not mean $g$ not differentiable in $0$ :
$$\frac{g(x)-g(0)}{x}=x\cos\frac{1}{x^2}\xrightarrow[x\to0]{}0$$
so $g'(0)=0$.
All you can say is that $g'$ is not continuous in $0$.
