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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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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![## Calculus: Finding the Derivative
### Problem Statement:
Find the derivative of the function:
\[ f(x) = \frac{\sec(x)}{\csc(x) + 3} \]
In this problem, we are looking to find the derivative of the given function \(f(x)\).
### Solution Method:
We will apply the quotient rule for differentiation. The quotient rule states:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
where \( u = \sec(x) \) and \( v = \csc(x) + 3 \).
1. First, we need to find the derivatives of \(u\) and \(v\):
- \( u = \sec(x) \)
- \( v = \csc(x) + 3 \)
2. Compute the derivatives:
- The derivative of \( u \) with respect to \( x \) is \( u' = \sec(x) \tan(x) \).
- The derivative of \( v \) with respect to \( x \) is \( v' = -\csc(x) \cot(x) \).
3. Applying the quotient rule:
- Substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula.
\[ f'(x) = \frac{\sec(x) \tan(x) (\csc(x) + 3) - \sec(x)(-\csc(x) \cot(x))}{(\csc(x) + 3)^2} \]
This completes the step of setting up the derivative using the quotient rule, and further simplification can be done to reach the final form of the derivative.
### Visual Explanation:
There are no graphs or diagrams included in this problem. The explanation relies purely on formula-based differentiation.
### Conclusion:
By correctly applying the quotient rule and substituting the corresponding values, you can find the derivative of the function \( f(x) = \frac{\sec(x)}{\csc(x) + 3} \). Further simplification helps to reach the most simplified form of the derivative if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75758824-d9ac-45ff-8935-fa266cdc3747%2Fd11e3d5c-acd2-40af-8719-30b9547392f6%2Fnohpi9l.png&w=3840&q=75)
Transcribed Image Text:## Calculus: Finding the Derivative
### Problem Statement:
Find the derivative of the function:
\[ f(x) = \frac{\sec(x)}{\csc(x) + 3} \]
In this problem, we are looking to find the derivative of the given function \(f(x)\).
### Solution Method:
We will apply the quotient rule for differentiation. The quotient rule states:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
where \( u = \sec(x) \) and \( v = \csc(x) + 3 \).
1. First, we need to find the derivatives of \(u\) and \(v\):
- \( u = \sec(x) \)
- \( v = \csc(x) + 3 \)
2. Compute the derivatives:
- The derivative of \( u \) with respect to \( x \) is \( u' = \sec(x) \tan(x) \).
- The derivative of \( v \) with respect to \( x \) is \( v' = -\csc(x) \cot(x) \).
3. Applying the quotient rule:
- Substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula.
\[ f'(x) = \frac{\sec(x) \tan(x) (\csc(x) + 3) - \sec(x)(-\csc(x) \cot(x))}{(\csc(x) + 3)^2} \]
This completes the step of setting up the derivative using the quotient rule, and further simplification can be done to reach the final form of the derivative.
### Visual Explanation:
There are no graphs or diagrams included in this problem. The explanation relies purely on formula-based differentiation.
### Conclusion:
By correctly applying the quotient rule and substituting the corresponding values, you can find the derivative of the function \( f(x) = \frac{\sec(x)}{\csc(x) + 3} \). Further simplification helps to reach the most simplified form of the derivative if needed.
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