The curve defined by x^ 2 3 +y^ 2 3 =8^ 2 3 is an astroid centered at the origin.

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

The curve defined by

x^{\tfrac{2}{3}}+y^{\tfrac{2}{3}}=8^{\tfrac{2}{3}}

is an astroid centered at the origin. (a) Differentiate the relation to obtain an explicit formula for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Determine all points on the astroid where the tangent line is vertical. Justify each coordinate pair you list. (c) Determine all points on the astroid where the tangent line is horizontal. Justify each coordinate pair you list. (d) Select one point from part (b) and write the equation of the corresponding vertical tangent line.

Related theory

Straight Lines
Advanced Differentiation
Implicit and parametric differentiation
Derivatives of implicit functions
Continuity and Differentiability

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