How to calculate the first and second order derivatives of the curve given in polar coordinates?

Question

How do I calculate the first and second order derivatives, $dy/dx$ and $d^2y/d^2x$, of the following curve given in polar coordinates, $r=cos(\theta)$?

I really have no idea where to start on this or how to solve this problem so any help leading me on the proper path is appreciated.

Answer 1

We have $x= r \cos(\theta)$ and $y = r \sin(\theta)$ Then $$r = \cos(\theta) \Leftrightarrow \sqrt{x^2 + y^2} = \frac{x}{\sqrt{x^2 + y^2}} \Leftrightarrow x^2 + y^2 = x$$ $y = \pm f(x)$ with $f(x) \equiv \sqrt{x(1-x)}$

And now you can derivate $$f'(x) = \frac{1}{2} (x(x-1))^{-1/2} (1-2x) $$

Answer 2

Hint:

$ r=\cos \theta$ is the circunference of center $(0.5,0)$ and diameter $1$. So you have: $$ y=\pm\sqrt{x-x^2} $$

can you derive?