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3 (8) Consider the curve given by 23 + x?y+ y³ = 1 Use implicit differentiation to find the equation of the tangent line to this curve at the point (3, –2). (9) Use logarithmic differentiation to find the derivative of psin: y = */1+x² (1+ e2=)³ You must use logarithmic differentiation correctly to get full credit.

3 (8) Consider the curve given by 23 + x?y+ y³ = 1 Use implicit differentiation to find the equation of the tangent line to this curve at the point (3, –2). (9) Use logarithmic differentiation to find the derivative of psin: y = */1+x² (1+ e2=)³ You must use logarithmic differentiation correctly to get full credit.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Implicit and Logarithmic Differentiation Problems #### Problem 8 Consider the curve given by: \[ x^3 + x^2y + y^3 = 1 \] - **Task**: Use implicit differentiation to find the equation of the tangent line to this curve at the point \( (3, -2) \). #### Problem 9 Use logarithmic differentiation to find the derivative of: \[ y = \frac{x^{\sin x} \sqrt{1 + x^2}}{(1 + e^{2x})^3} \] - **Note**: You must use logarithmic differentiation correctly to get full credit. ### Explanation **Implicit Differentiation**: - A technique used when differentiating equations not expressed explicitly in terms of one variable. It is especially useful for curves described by an equation involving the variables in a complicated way. **Logarithmic Differentiation**: - A method that simplifies the differentiation of expressions involving products, quotients, or powers by taking the natural logarithm of both sides before differentiating.
Transcribed Image Text:### Implicit and Logarithmic Differentiation Problems #### Problem 8 Consider the curve given by: \[ x^3 + x^2y + y^3 = 1 \] - **Task**: Use implicit differentiation to find the equation of the tangent line to this curve at the point \( (3, -2) \). #### Problem 9 Use logarithmic differentiation to find the derivative of: \[ y = \frac{x^{\sin x} \sqrt{1 + x^2}}{(1 + e^{2x})^3} \] - **Note**: You must use logarithmic differentiation correctly to get full credit. ### Explanation **Implicit Differentiation**: - A technique used when differentiating equations not expressed explicitly in terms of one variable. It is especially useful for curves described by an equation involving the variables in a complicated way. **Logarithmic Differentiation**: - A method that simplifies the differentiation of expressions involving products, quotients, or powers by taking the natural logarithm of both sides before differentiating.
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