derivatives - Differentiating polar functions using complex numbers

I was wondering, given some polar function $r(\theta)$ is it possible to convert it into a complex number in exponential form, then differentiate that and then convert it back and have the appropriate derivative of the polar function?

For example take the polar function $r=\cos(a\theta)$, also known as a rose curve for $a\in\mathbb{Q}$. Is it possible to 'complexify' this function (not too sure how possible that is) and then take the derivative?