
We differentiate polar functions.

The previous section discussed a special class of parametric functions called polar functions. We know that

and so we can compute the derivative of $y$ with respect to $x$ using differentials: provided that $x'(t) \ne 0$. With polar functions we have

so

provided that $x'(\theta )\ne 0$.

When the graph of the polar function $r(\theta )$ intersects the origin (sometimes called the “pole”), then $r(\alpha )=0$ for some angle $\alpha$.

When $r(\alpha ) = 0$, what is the formula for $\dd [y]{x}$?
$\dd [y]{x}= \sin (\alpha )$. $\dd [y]{x}= \cos (\alpha )$. $\dd [y]{x}= \tan (\alpha )$.
The answer to the question above leads us to an interesting point. It tells us the slope of the tangent line at the pole. When a polar graph touches the pole at $\theta =\alpha$, the equation of the tangent line in polar coordinates at the pole is $\theta =\alpha$.