9.7 Defining Polar Coordinates and Differentiating in Polar Form

Part of Parametric Equations, Polar Coordinates, and Vector-Valued Functions in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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In polar coordinates, a point is described by (r,θ) instead of (x,y). Here r is the distance from the origin and θ is the angle from the positive x-axis. The rectangular and polar systems are connected by

{x=r\cos\theta}

and

{y=r\sin\theta}

If r is given as a function of θ, then you can differentiate parametrically by treating θ as the parameter. The slope formula is

\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}

provided

{\frac{dx}{d\theta}\ne0.}

You should think of polar differentiation as a parametric problem in disguise. Once x and y are rewritten in terms of ${\theta}$ , the same derivative machinery from parametric equations applies. Polar graphs can create roses, limacons, and spirals, so this system describes curves that are often awkward in rectangular form. The main challenge is converting correctly and differentiating carefully.

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