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Derivatives of Polar Equations for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: More Applications of Derivatives Practice Problems for AP Calculus

For polar representations, remember that r = f(θ), so x = r cos θ = f (θ) cos θ and y = r sin θ = f (θ) sin θ. Differentiating with respect to θ requires the product rule.

Example

Find the equation of the tangent line to the curve r =2+2 sin θ when θ =

Step 1:   = 2 cos θ

Step 2:

    = – (2 + 2 sin θ) sin θ + cos θ(2 cos θ) = 2(cos2θ – sin2θ – sin θ).
    By the Pythagorean identity,
    2(cos2θ – sin2θ – sin θ)=2(1 – sin2θ – sin2θ – sin θ)
    = 2(1 – sinθ – 2sin2θ = 2(1 - 2 sinθ)(1 + sinθ).
    Also, = (2 + 2 sin θ) cosθ + sinθ(2 cosθ(1 + 2 sin θ).

Step 3:  

Step 4:  

Step 5:   When

Step 6:   The equation of the tangent line is

Practice problems for these concepts can be found at: More Applications of Derivatives Practice Problems for AP Calculus

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