Show that $f$ is differentiable on $\mathbb{R}$ and give the value of $f'(x)$ for all $x$ in $\mathbb{R};$ justify any assertions you make. $$f(x)=\begin{cases}x^2\sin\left(\frac{1}{x^2}\right)\cos(x^4),\quad > &x\not=0 \\ 0, &x=0\end{cases}. $$
How do I go about this?
HINT
For $x\neq 0$ we can calculate directly $f'(x)$.
For $x=0$ let consider the definition
$$f'(0)=\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0}\,x \sin\left(\frac{1}{x^2}\right)\cos(x^4)$$