Uniqueness of a weak derivative

Question

Let $f\in \mathrm{L}^2[a,b]$. As usually $f'$ is the so called weak derivative of $f$ if $\forall \phi \in C_c^{\infty}(a,b)$ $\int_a^b f'\phi dt=-\int_a^bf \phi' dt$. Is it reasonably to think that $f'$ is unique, i.e. all weak derivatives of $f$ are equivalent. Could you help me to prove it?