fa.functional analysis – Decay estimate of Fourier transform of a compactly supported function

Assume $f(x), x in mathbb{R}$ is a function with a compact support such that its Fourier transform $hat{f}(xi)$ has a decay rate
$$hat{f}(xi) lesssim frac{1}{|xi|^gamma + 1}$$
for some $gamma ge 1$.
Now set $$h(x) = xf(x).$$ Since $f$ has a compact support, $h$ should have similar or better regularity than $f$. Can we now get the following decay estimate of the Fourier transform of $h$ ?
$$hat{h}(xi) lesssim frac{1}{|xi|^gamma + 1}$$
I know now we have $hat{h}(xi) = -ipartial_xi hat{f}(xi)$, but it seems hard to only use this relation to get the decay estimate.