How to build $f$ differentiable on $[0,1]$ easily, such that $f'$ is not Riemann-integrable?

Question

How to build $f:[0,1]\to \mathbb{R}$ differentiable on $[0,1]$ easily, such that $f'$ is not Riemann-integrable?

Answer 1

Consider $f(x) = x^2\sin(1/x^2)$ for $x>0$ and $f(0)= 0$. You can check that this is differentiable but its derivative is unbounded and so cannot be Riemann integrable.