I have two Time Domain functions, $f_1(t)$ and $f_2(t)$.

I have both Fourier Transforms, $F_1(\omega)$ and $F_2(\omega)$. Functions $f_1$ and $f_2$ are not independent and, in fact, $f_1$ is also a function of $f_2$. So, the derivative $f_3(t)=\frac{d\,f_1}{df_2\,}$ is meaningful.

Can I obtain the Fourier Transform $F_3(\omega)$ (the FT of $f_3$) directly from $F_1(\omega)$ and $F_2(\omega)$?

I wish to calculate it directly in the Frequency Domain, without going to the Time Domain to do the derivative.

  • $\begingroup$ I'm tired, so I'll ask this rather than prove it from hand: I think it makes a difference whether you use $\frac{d f}{dt}$ or $\frac{\partial f}{\partial t}$. You definitely mean the total derivative, right? $\endgroup$ Jan 15 '18 at 19:51
  • $\begingroup$ Is there a way to know this without more information? What if ${f_3}(t) = {f_1}(t)^{{f_2}(t)}$? $\endgroup$ Jan 16 '18 at 3:25

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