Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $$x^{2}+x y+y^{2}=3$$ (ellipse)

Name: Problem 23, Chapter 3: Calculus
Uploaded: 2023-03-14T01:22:35-08:00
Channel: Nicole Hoffman
Description: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^2+x y+y^2=3 (ellipse)

   Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$$x^{2}+x y+y^{2}=3$$ (ellipse)

Calculus

James Stewart 4th Edition

Chapter 3, Problem 23

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Solved by Expert Nicole Hoffman

03/14/2023

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Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $$x^{2}+x y+y^{2}=3$$ (ellipse)

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Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated on one side. Instead of solving for the dependent variable explicitly, both sides of the equation are differentiated with respect to the independent variable, using the chain rule as necessary.

Finding the Slope of a Tangent Line

To determine the slope of a tangent line to a curve defined implicitly, differentiate the equation of the curve implicitly to obtain an expression for the derivative dy/dx. This derivative represents the slope of the tangent at any point on the curve.

Point-Slope Form for Tangent Lines

Once the slope at a specific point on the curve is found, the point-slope form of the line, y - y1 = m(x - x1), is used to write the equation of the tangent line. Here, m is the slope from the derivative and (x1, y1) is the given point on the curve.

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