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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![### Calculus Problem - Finding the derivative of the inverse function
**Problem Statement:**
For the function \( y = f(x) = x^2 - 2x \), where \( x \geq 1 \), find:
\[
\left. \frac{d f^{-1}}{dy} \right|_{y=3}
\]
**Solution:**
### Finding the Derivative:
We need to find \( \left( f^{-1} \right)'(3) \), which represents the derivative of the inverse function evaluated at \( y=3 \).
1. **Inverse Function Derivative Formula**:
The derivative of the inverse function \( f^{-1}(x) \) at the point \( y \) is given by:
\[
\left( f^{-1} \right)'(y) = \frac{1}{f'(f^{-1}(y))}
\]
2. **Evaluate \( f^{-1}(3) \)**:
We need to find the \( x \) value such that \( f(x) = 3 \):
\[
x^2 - 2x = 3
\]
Rearrange and solve for \( x \):
\[
x^2 - 2x - 3 = 0
\]
Factoring the quadratic equation:
\[
(x-3)(x+1) = 0
\]
Therefore, \( x = 3 \) or \( x = -1 \). Given the constraint \( x \geq 1 \), we select \( x = 3 \).
3. **Find \( f'(x) \)**:
Compute the derivative \( f'(x) \) of \( f(x) \):
\[
f(x) = x^2 - 2x
\]
Differentiate with respect to \( x \):
\[
f'(x) = 2x - 2
\]
Evaluate \( f'(x) \) at \( x = 3 \):
\[
f'(3) = 2(3) - 2 = 6 - 2 = 4
\]
4. **Substitute in the Inverse Derivative Formula**:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2F707fd625-c5f3-4110-9887-33d892b78f14%2Fr0t70bc_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem - Finding the derivative of the inverse function
**Problem Statement:**
For the function \( y = f(x) = x^2 - 2x \), where \( x \geq 1 \), find:
\[
\left. \frac{d f^{-1}}{dy} \right|_{y=3}
\]
**Solution:**
### Finding the Derivative:
We need to find \( \left( f^{-1} \right)'(3) \), which represents the derivative of the inverse function evaluated at \( y=3 \).
1. **Inverse Function Derivative Formula**:
The derivative of the inverse function \( f^{-1}(x) \) at the point \( y \) is given by:
\[
\left( f^{-1} \right)'(y) = \frac{1}{f'(f^{-1}(y))}
\]
2. **Evaluate \( f^{-1}(3) \)**:
We need to find the \( x \) value such that \( f(x) = 3 \):
\[
x^2 - 2x = 3
\]
Rearrange and solve for \( x \):
\[
x^2 - 2x - 3 = 0
\]
Factoring the quadratic equation:
\[
(x-3)(x+1) = 0
\]
Therefore, \( x = 3 \) or \( x = -1 \). Given the constraint \( x \geq 1 \), we select \( x = 3 \).
3. **Find \( f'(x) \)**:
Compute the derivative \( f'(x) \) of \( f(x) \):
\[
f(x) = x^2 - 2x
\]
Differentiate with respect to \( x \):
\[
f'(x) = 2x - 2
\]
Evaluate \( f'(x) \) at \( x = 3 \):
\[
f'(3) = 2(3) - 2 = 6 - 2 = 4
\]
4. **Substitute in the Inverse Derivative Formula**:
\[
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