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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1300845a-828a-484d-8f06-319293900125%2F0bc88ac5-6979-4508-bd25-efa699d3a0c0%2Fm1f34vv_processed.png&w=3840&q=75)
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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