The curve is defined implicitly by the equation .

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

Problem

The curve

C

is defined implicitly by the equation

e^{xy}+y=x^{2}

. (1) Derive an explicit formula for

\dfrac{dy}{dx}

in terms of

x

and

y

. Present your work in a logically ordered sequence of steps. (2) Determine all points on

C

where the tangent line is horizontal. Express each point’s coordinates as ordered pairs and justify that your list is complete. (3) For

x>0

, decide whether the curve can possess a vertical tangent. Provide a mathematical argument that supports your conclusion.

Related theory

Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of implicit functions
Continuity and Differentiability

Browse the problem library