Homework Help: 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.

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