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Let be $f\in L^1(\mathbb{R})$,

I will be able to say that

$$ \dfrac{\hat{df(w)}}{dx} = \int_{-\infty}^{\infty}\dfrac{df(x)}{dx}\exp(-2\pi j wx)dx $$?

Why?

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Are you trying to differentiate a function of $w$ with respect to $x$? That doesn't look very promising. Also, if you are asking whether the derivative of the Fourier transform is the Fourier transform of the derivative, that is wrong in general. – Lukas Geyer Oct 29 '12 at 4:17

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