The relation defines implicitly as a differentiable function of near the point .

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

Problem

The relation

x^{2}+xy+y^{2}=7

defines

y

implicitly as a differentiable function of

x

near the point

(1,2)

. (1) Obtain an expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. Simplify as much as possible. (2) Differentiate your result from part (a) with respect to

x

to obtain an explicit formula for

\dfrac{d^{2}y}{dx^{2}}

in terms of

x

and

y

. Show all algebraic steps. (3) Evaluate

\dfrac{d^{2}y}{dx^{2}}

at the point

(1,2)

and state whether the curve is concave upward or concave downward there. Justify your conclusion.

Related theory

Higher-Order Derivatives
Advanced Differentiation
Implicit and parametric differentiation
Derivatives of implicit functions
Continuity and Differentiability

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