$$ f(t)=\frac{\sin(at)}{t} $$
Since the term is parameterized, it's easy to see that if I take the first derivative with respect to 'a', then the function becomes considerably easier. I do this to the Fourier Transform and obtain: $$ \frac{\partial }{\partial a}\Im (f(t))=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }\cos(at))e^{itx}dt $$
However, this is an integral of an even function times an odd function, which equals 0 and raises my suspicion. I've tried implementing Euler's cosine form and got nowhere.
Also I'm using the imaginary symbol as the Fourier transform. Why? It looks cool.
The differentiating inside the integral trick requires several conditions be checked first. If you notice, the integral on the right is not even defined.
Check out http://ocw.mit.edu/courses/mathematics/18-304-undergraduate-seminar-in-discrete-mathematics-spring-2006/projects/integratnfeynman.pdf
and The Integral that Stumped Feynman?
Also you have to be a bit careful with how you're defining everything. You should call the Fourier transform $\hat{f}(x)$ rather than $f(t)$ since it is a different function in the variable $x$.