Calculus/Polar Differentiation

Differential calculus

We have the following formulas:

$r\frac{\partial }{\partial r}=x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}$

$\frac{\partial }{\partial \theta }=-y\frac{\partial }{\partial x}+x\frac{\partial }{\partial y}.$

To find the Cartesian slope of the tangent line to a polar curve r(θ) at any given point, the curve is first expressed as a system of parametric equations.

$x=r(\theta )\cos \theta$

$y=r(\theta )\sin \theta$

Differentiating both equations with respect to θ yields

$\frac{\partial x}{\partial \theta }=r^{\prime }(\theta )\cos \theta -r(\theta )\sin \theta$

$\frac{\partial y}{\partial \theta }=r^{\prime }(\theta )\sin \theta +r(\theta )\cos \theta$

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r, r(θ)):

$\frac{dy}{dx}=\frac{r^{\prime }(\theta )\sin \theta +r(\theta )\cos \theta }{r^{\prime }(\theta )\cos \theta -r(\theta )\sin \theta }$