I was looking for an earliest reference or the name of the mathematician who showed calculating the derivatives is possible in the Fourier domain?


The Fourier transform of the derivative is ([Wikipedia](https://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships))
$$
\mathcal{F}(f')(\xi)=2\pi i\xi\cdot\mathcal{F}(f)(\xi).
$$


Thank you.



Earliest reference on the calculation of derivatives by Fourier transform
Despite this being complicated by Fourier’s use of sine and cosine transforms, I think @PiyushGrover is right: Théorie analytique de la chaleur, [Art. 419](//archive.org/details/thorieanalytiq00four/page/557):



> 
> $$
> \begin{align}
> \frac{d^{2i}}{dx^{2i}}fx
> =\pm&\int d\alpha f\alpha\int dp\,p^{2i}\cos.(px-p\alpha)\\
> \frac{d^{(2i+1)}}{dx^{2i+1}}fx
> =\mp\frac1{2\pi}&\int d\alpha f\alpha\int dp\,p^{2i+1}\sin.(px-p\alpha)
> \end{align}
> $$
> 
> 
> 


and Art. 422 “Expression générale de la fluxion d’ordre $i$”.


