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 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!