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I'm stuck on trying to find the fourier transform of $te^{-a|t|}$ without integration. I have to use the fact that the fourier transform of $e^{-a|t|}$ is $2a / (a^2 + w^2)$ in my calculation. This should be a problem regarding properties of fourier transforms but integrals and derivatives don't seem to help and I'm not sure where to go from here.

Nevermind, figured it out. Use the property $F\{x(t)\} = X(jw)$, $F\{tx(t)\} = j\frac{d}{dw}X(jw)$

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The result is straightforward by using the property $$\frac{d}{{dw}}F(f(t))(w) = - F(tf(t))(w)$$.

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