Clearly, this is not fun.

If we decompose f(z) and g(z) into real valued functions u + iv and s + it, it seems like a reasonable first step to multiply the numerator and denominator by the complex conjugate of g (s - it), which yields something that looks very similar to the desired result. However, when we take the derivative of these real valued functions, the denominator is no longer $\displaystyle g^2$ and the numerator looks more like f' multiplied by the complex conjugate of g, minus g'f.

Any hints on better steps to take?

Thank you!