Fourier transform of the derivative - insufficient hypotheses? An exercise in Carlos ISNARD's Introdução à medida e integração:

Show that if $f$ and $f'$ $\in\mathscr{L}^1(\mathbb{R},\lambda,\mathbb{C})$ and $\lim_{x\to\pm\infty}f(x)=0$ then $\hat{(f')}(\zeta)=i\zeta\hat{f}(\zeta)$.

($\lambda$ is the Lebesgue measure on $\mathbb{R}$.)
I'm tempted to apply integration by parts on the integral from $-N$ to $N$ and then take limit as $N\to\infty$. But to obtain the result I seemingly need $f'e^{-i\zeta x}$ to be Riemann-integrable so as to use the fundamental theorem of Calculus.
What am I missing here?
Thank you.

There is a measure-theoretic version of the fundamental theorem of Calculus, see here.
