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...
Related questions
Question
Finds the derivative of the given function.
![**Expression (k)**
The expression given is:
\[ z = \ln \left( \sqrt{u^2 + 25} - u \right) \]
**Explanation:**
- **\( z \)**: Represents the variable that is dependent on the expression within the natural logarithm, \(\ln\).
- **\(\ln\)**: Denotes the natural logarithm, which is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828.
- **\(\sqrt{u^2 + 25}\)**: Represents the square root of the sum of \( u \) squared and 25.
- **\(- u\)**: Means that the variable \( u \) is subtracted from the square root term.
This expression involves both logarithmic and square root functions and is used in contexts where the rate of change of a quantity is described by its proportional relationship to another variable. Understanding how to manipulate this expression is important in calculus and higher-level mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3651bc5-cd2c-43d5-bdfb-11943a13f854%2Fcfde5a4c-a44b-4913-b705-0bf5960983f4%2Fbm3bsh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Expression (k)**
The expression given is:
\[ z = \ln \left( \sqrt{u^2 + 25} - u \right) \]
**Explanation:**
- **\( z \)**: Represents the variable that is dependent on the expression within the natural logarithm, \(\ln\).
- **\(\ln\)**: Denotes the natural logarithm, which is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828.
- **\(\sqrt{u^2 + 25}\)**: Represents the square root of the sum of \( u \) squared and 25.
- **\(- u\)**: Means that the variable \( u \) is subtracted from the square root term.
This expression involves both logarithmic and square root functions and is used in contexts where the rate of change of a quantity is described by its proportional relationship to another variable. Understanding how to manipulate this expression is important in calculus and higher-level mathematics.
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