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The image contains a handwritten mathematical expression followed by a breakdown of its derivatives, commonly used in calculus. ### Original Expression: 1. \( x^2 (\sqrt{x-2})(1+x^2)^5 (2-x)^4 \) ### Derivatives: - The expression is expanded using logarithmic differentiation as follows: #### Logarithmic Components: - \( 2m(x) \) - \( 1m(x^2) \ln(\sqrt{x-2}) \) - \( 5 \ln(1+x^2) \) - \( 4 \ln(2-x) \) ### Derivative Breakdown: 1. \( 2 \cdot \frac{1}{x} \) 2. \( + \frac{1}{\sqrt{x-2}} \cdot \left( \frac{1}{2\sqrt{x}} \right) \) 3. \( + 5 \cdot \frac{1}{1+x^2} \cdot (2x) \) 4. \( + 4 \cdot \frac{1}{2-x} \cdot (-1) \) This expression and its derivatives could be used in an educational setting to illustrate principles such as product rule, chain rule, and logarithmic differentiation in calculus.

The image contains a handwritten mathematical expression followed by a breakdown of its derivatives, commonly used in calculus. ### Original Expression: 1. \( x^2 (\sqrt{x-2})(1+x^2)^5 (2-x)^4 \) ### Derivatives: - The expression is expanded using logarithmic differentiation as follows: #### Logarithmic Components: - \( 2m(x) \) - \( 1m(x^2) \ln(\sqrt{x-2}) \) - \( 5 \ln(1+x^2) \) - \( 4 \ln(2-x) \) ### Derivative Breakdown: 1. \( 2 \cdot \frac{1}{x} \) 2. \( + \frac{1}{\sqrt{x-2}} \cdot \left( \frac{1}{2\sqrt{x}} \right) \) 3. \( + 5 \cdot \frac{1}{1+x^2} \cdot (2x) \) 4. \( + 4 \cdot \frac{1}{2-x} \cdot (-1) \) This expression and its derivatives could be used in an educational setting to illustrate principles such as product rule, chain rule, and logarithmic differentiation in calculus.

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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The image contains a handwritten mathematical expression followed by a breakdown of its derivatives, commonly used in calculus. ### Original Expression: 1. \( x^2 (\sqrt{x-2})(1+x^2)^5 (2-x)^4 \) ### Derivatives: - The expression is expanded using logarithmic differentiation as follows: #### Logarithmic Components: - \( 2m(x) \) - \( 1m(x^2) \ln(\sqrt{x-2}) \) - \( 5 \ln(1+x^2) \) - \( 4 \ln(2-x) \) ### Derivative Breakdown: 1. \( 2 \cdot \frac{1}{x} \) 2. \( + \frac{1}{\sqrt{x-2}} \cdot \left( \frac{1}{2\sqrt{x}} \right) \) 3. \( + 5 \cdot \frac{1}{1+x^2} \cdot (2x) \) 4. \( + 4 \cdot \frac{1}{2-x} \cdot (-1) \) This expression and its derivatives could be used in an educational setting to illustrate principles such as product rule, chain rule, and logarithmic differentiation in calculus.
Transcribed Image Text:The image contains a handwritten mathematical expression followed by a breakdown of its derivatives, commonly used in calculus. ### Original Expression: 1. \( x^2 (\sqrt{x-2})(1+x^2)^5 (2-x)^4 \) ### Derivatives: - The expression is expanded using logarithmic differentiation as follows: #### Logarithmic Components: - \( 2m(x) \) - \( 1m(x^2) \ln(\sqrt{x-2}) \) - \( 5 \ln(1+x^2) \) - \( 4 \ln(2-x) \) ### Derivative Breakdown: 1. \( 2 \cdot \frac{1}{x} \) 2. \( + \frac{1}{\sqrt{x-2}} \cdot \left( \frac{1}{2\sqrt{x}} \right) \) 3. \( + 5 \cdot \frac{1}{1+x^2} \cdot (2x) \) 4. \( + 4 \cdot \frac{1}{2-x} \cdot (-1) \) This expression and its derivatives could be used in an educational setting to illustrate principles such as product rule, chain rule, and logarithmic differentiation in calculus.
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