Use the definition of differentiation on a piecewise function. I need to find the derivative at $x=0$.
$$ f(x)= \begin{cases} x^2\sin(1/x) & \text{if } x\neq 0 \\ 
0 & \text{if } x \leqslant 0 \end{cases} $$
Using the definition, I know that it's equal to $0$. However, I also need to prove that $f'(x)$ is not continuous at $x=0$. Do I need the entire equation for that?

Indeed $f'(0) = 0$. We have, then: $$f'(x) = \begin{cases} 2x\sin(1/x) - \cos(1/x),& \text{if }x\neq 0 \\ 0, & \text{if } x \leq 0\end{cases}$$
Just check that $\lim_{x \to 0}f'(x)$ does not exist and you are done.
