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.
Note that your function $g(x)$ is not defined at $x=0$, therefore neither is its derivative. A necessary condition for the derivative to exist at a point is to the function be defined and to be continuos at that point, which is not 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 $0$.