Derivatives in Polar Coordinates

AP Calculus · Axiom Academy

LESSON Derivatives in Polar Coordinates Finding slopes of tangent lines to polar curves Step 1: Polar Curves Are Parametric! Here's a key insight: every polar curve is actually a parametric curve in disguise. If , we can write: The parameter is θ! So we can use the parametric derivative formula we already know! Step 2: Polar Derivative Formula Since polar curves are parametric with parameter θ, we use the parametric derivative formula: Let's compute the derivatives. We have and . Step 3: Simplified Polar Derivative After computing and simplifying (using the quotient rule and product rule), we get: This is the formula you'll use! It's derived from the parametric derivative but is much more compact. dr/dθ is the derivative of r with respect to θ The numerator has the term "r dθ" which comes from the product rule This works for any polar curve! This makes sense! At the top of the circle (θ = π/2), the slope is 0. At the side (θ = 0), the slope is undefined (vertical). Step 5: Finding Horizontal and Vertical Tangents Horizontal Tangents: Occur where Solve r'cos(θ) + r sin(θ) = 0 for θ Vertical Tangents: Occur where Solve r'sin(θ) - r cos(θ) = 0 for θ These are useful for finding extrema and special points on polar curves!

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