I am trying to solve a PDE using Fourier transformation similar to how the telegraph equation might be solved. This is done by Fourier transforming each term. It happens that one of the terms is given by
$$f(x)\frac{d\,f(x)}{dx}\frac{d\,g(x)}{dx}.$$
I am trying to take the Fourier transform of this, which would boil down to the integral
$$\int_{-\infty}^{\infty}f(x)\frac{d\,f(x)}{dx}\frac{d\,g(x)}{dx} e^{-ikx} dx.$$
I'm not sure how to approach this problem. I know the general property
$$\text{FT}[f^{(n)}(x)]=(ik)^n\,\hat{f}(k)$$
for a derivative but I have never encountered a Fourier transform that is a product of several functions and derivatives. Does anyone know a trick, theorem or hint that could point me in the right direction? Or would this be very difficult to simplify? I have limited experience with the Fourier transformation and I am looking to learn.