Textbook Problem

Curvature A curve C is given by the polar equation $r=f(\theta )$. Show that the curvature K at the point $(r,\theta )$ is

$K=\frac{\left|2{(r^{\text{'}})}^2-r{r}^{"}+r^2\right|}{{[{(r^{\text{'}})}^2+r^2]}^{3/2}}$

[Hint: Represent the curve by $r(\theta )=r\cos \theta i+r\sin \theta j$.]