Differentiate Archimedes's spiral I read that the only problem of differential calculus Archimedes solved was constructing the tangent to his spiral,
$$r = a + b\theta$$
I would like to differentiate it but I don't know much about differentiating polar functions and can't find this particular problem online. Without giving me a full course in differential geometry, how does one calculate the tangent to the curve at $\theta$?

This is much easier to swallow in complex variables. Let
$$z=(a+b\theta)e^{i\theta}$$
then
$$\dot z=[(a+b\theta)i+b]e^{i\theta}=[(a+b\theta)i+b][\text{cos}\theta + i\text{sin}\theta]$$
from which the previous results for $\dot x$ and $\dot y$ can be readily obtained. However, we don't need that to get the tangent, because it is, in fact, just $\dot z$, with the caveat that you need to express it as an angle, let's call it $\Theta$. Thus,
$$\Theta=\text{tan}^{-1}\frac{\mathfrak{Im}(\dot z)}{\mathfrak{Re}(\dot z)}$$
