Inverse discrete time Fourier transform with differentiation Consider a signal x[n] and its DTFT $X(e^{jω})$ . Assume $X(e^{jω})$ is differentiable. Compute the inverse DTFT of
$j\frac{dX(e^{jω})}{d\omega}$
You should write your answer in terms of $x[n]$ and elementary functions and constants, for example $π/2x[n]$ would be written :
$\pi / 2x[n]$
Hint:
differentiate from Fourier Transform equation:
$$X(e^{j\omega})= \sum\limits_{n=- \infty }^{\infty}x[n]e^{-jn\omega} \Rightarrow \frac{d}{d\omega}X(e^{j\omega})= \frac{d}{d\omega}\sum\limits_{n=- \infty }^{\infty}x[n]e^{-jn\omega}$$
and then find what you want.