The astroid is described by the equation with and real numbers.

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

The astroid is described by the equation

x^{2/3}+y^{2/3}=4^{2/3}

with

x

and

y

real numbers. (a) Find

\dfrac{dy}{dx}

for the astroid. (b) Determine all points on the astroid where the tangent line is vertical. (c) Identify the coordinates of any points at which both the numerator and denominator of

\dfrac{dy}{dx}

are zero, and explain whether these points correspond to cusps or smooth vertical tangents. (d) For one point identified in part (b), write the equation of the corresponding vertical tangent line.

Related theory

Non-Linear Equations
Advanced Differentiation
Implicit and parametric differentiation
Derivatives of implicit functions
Continuity and Differentiability

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