# Properties of the Fourier Transform - Time Differentitation

1. Aug 8, 2012

### p75213

1. The problem statement, all variables and given/known data
This is copied from a book:
\eqalign{ & {\rm{Time Differentitation}} \cr & {\rm{Given that: }}F(\omega ) = F\left[ {f(t)} \right] \cr & F\left[ {f'(t)} \right] = jwF(\omega ) \cr & {\rm{Proof:}} \cr & f(t) = {F^{ - 1}}\left[ {F\left( \omega \right)} \right] = {1 \over {2\pi }}\int_{ - \infty }^\infty {F\left( \omega \right){e^{j\omega t}}d\omega } \cr & {\rm{Taking the derivative of both sides with respect to }}t{\rm{ gives:}} \cr & {d \over {dt}}f(t) = {{j\omega } \over {2\pi }}\int_{ - \infty }^\infty {F\left( \omega \right){e^{j\omega t}}d\omega } = j\omega {F^{ - 1}}\left[ {F(\omega )} \right]{\rm{ or }}F\left[ {f'(t)} \right] = jwF(\omega ) \cr}

Can somebody explain why the jw is outside the integral? I can't see how that happens using Leibniz's integral rule - http://en.wikipedia.org/wiki/Leibniz_integral_rule

2. Relevant equations

3. The attempt at a solution

2. Aug 8, 2012

### marcusl

The factor of jw should be inside the integral, as you surmise. Then take the forward FT of both sides to get the answer.

3. Aug 9, 2012

### p75213

Thanks marcusl. I see now.