Implicit differentiation Use Theorem 15.9 to evaluate dy/dx. Assume each equation implicitly defines y as a differentiable function of x. THEOREM 15.9 Implicit DifferentiationLet F be differentiable on its domain and suppose F(x, y) = 0 defines y as a differ-entiable function of x. Provided F, + 0,dyF.dxF,|| Vr? + 2ry + y4 = 3

The given equation is x2+2xy+y4=3. Consider Fx, y=x2+2xy+y4-3. This implies that, Fx=12x2+2xy+y4…

Answered: THEOREM 15.9 Implicit Differentiation…

THEOREM 15.9 Implicit Differentiation Let F be differentiable on its domain and suppose F(x, y) = 0 defines y as a differ- entiable function of x. Provided F, + 0, dy F. dx F, || Vr? + 2ry + y4 = 3

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.4: Derivatives Of Exponential Functions

Problem 37E: Use graphical differentiation to verify that ddxex=ex.

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Question

Implicit differentiation Use Theorem 15.9 to evaluate dy/dx. Assume each equation implicitly defines y as a differentiable function of x.

THEOREM 15.9 Implicit Differentiation
Let F be differentiable on its domain and suppose F(x, y) = 0 defines y as a differ-
entiable function of x. Provided F, + 0,
dy
F.
dx
F,
||

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

Expert Solution

Step 1

The given equation is $\sqrt{x^{2} + 2 x y + y^{4}} = 3$ .

Consider $F (x, y) = \sqrt{x^{2} + 2 x y + y^{4}} - 3$ .

This implies that,

$F_{x} = \frac{1}{2 \sqrt{x^{2} + 2 x y + y^{4}}} (2 x + 2 y) = \frac{x + y}{\sqrt{x^{2} + 2 x y + y^{4}}} F_{y} = \frac{1}{2 \sqrt{x^{2} + 2 x y + y^{4}}} (2 x + 4 y^{3}) = \frac{x + 2 y^{3}}{\sqrt{x^{2} + 2 x y + y^{4}}}$

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