I have somme problems understanding the following definition of distributional derivative.
I'm aware that to reach this definition integration by part is used and that for $f\in \mathrm{C^{1}}$, $$ \int_{-\infty}^{\infty}f'(x)\,\phi(x)\,\mathrm dx=-\int_{-\infty}^{\infty} f(x)\, \phi'(x) \,\mathrm dx $$
My question is, since we assumed $f$ to be differentiable to obtain this result, why can we apply the latter integral even to non differentiable functions like Heaviside step function or Dirac distribution? Shouldn't those function be differentiable too to apply the definition of distributional derivative or there's some flaw in my logic that I fail to see?
The integral notation in this case is an abuse of notation. To be more rigorous, one makes the difference between the integral of the multiplication of two functions $$ \int f \,g $$ and the evaluation of a distribution $f$ at some test function $g$, sometimes denoted by $$ \langle f,g\rangle. $$ In the case when $f$ is locally integrable, then the action of the corresponding distribution is just $$ \langle f,g\rangle = \int f\,g. $$ Then, as written in the comments, to mimic and be compatible with the integration by parts formula in the case of two sufficiently regular functions, one defines the distributional derivative of a distribution $f$ by the formula $$ \langle f',g\rangle =-\langle f,g'\rangle. $$ for any nice test function $g$. In the case when $f'$ is locally integrable, one recovers the classical integration by parts formula. In the other cases, writing the above formula with integrals is a (convenient) abuse of notation.