9.7 Defining Polar Coordinates and Differentiating in Polar Form

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Polar functions are functions that are graphed around a pole in a circular system rather than the Cartesian rectangular system. Polar functions are graphed with the points (r, θ) rather than (x, y).

When we are working with polar graphs, we can’t differentiate them right away. We have to convert them to Cartesian graphs. Converting polar equations to Cartesian also helps us visualize them. 

When we take derivatives of polar functions, we can take them as dr/dθ, which would give us the points that are furthest away from the origin on the polar coordinate system. We find dr/dθ in the same way we would find any normal derivative: by taking the derivative of the polar function: 

While dr/dθ can tell us relative maximum and minimum values, it doesn’t tell us the slope of the tangent line, since we can’t have linear graphs on the polar coordinate system. In order to find the slope of the tangent line, we need to find the derivative on the Cartesian system, which requires us to calculate dy/dx.

Of course, you can memorize this formula, but most students find it much easier to simply derive it using the chain rule.