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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![### Problem Statement
Given the equation:
\[ f(x) + x^2[f(x)]^3 = 10 \]
and
\[ f(1) = 2 \]
find \( f'(1) \).
### Detailed Explanation
We are provided with a functional equation involving \( f(x) \) and an initial condition for \( f \). Our goal is to find the first derivative of the function \( f \) at \( x = 1 \).
To solve this, we will:
1. Differentiate both sides of the given equation implicitly with respect to \( x \).
2. Substitute \( x = 1 \) and \( f(1) = 2 \) into the resulting equation.
3. Solve for \( f'(1) \).
### Solution Steps:
1. **Differentiate both sides of the equation implicitly:**
The given equation is:
\[ f(x) + x^2[f(x)]^3 = 10 \]
Taking the derivative with respect to \( x \), we apply the chain rule:
\[ \frac{d}{dx}\left[ f(x) \right] + \frac{d}{dx} \left[ x^2 (f(x))^3 \right] = \frac{d}{dx} [10] \]
Since the derivative of a constant is zero:
\[ f'(x) + \frac{d}{dx} \left[ x^2 (f(x))^3 \right] = 0 \]
2. **Differentiate the product \( x^2 (f(x))^3 \) using the product rule:**
Let \( u = x^2 \) and \( v = (f(x))^3 \). Then:
\[ \frac{d}{dx} [uv] = u'v + uv' \]
Here, \( u' = 2x \) and \( v = (f(x))^3 \):
\[ \frac{d}{dx} \left[ x^2 (f(x))^3 \right] = 2x (f(x))^3 + x^2 \left[ 3 (f(x))^2 f'(x) \right] \]
3. **Combine the derivatives:**
\[ f'(x) + 2x (f(x))^3 + x^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffc15f15-a21f-4d69-ba6d-8e9396a0e005%2F749f089e-fec1-49a4-8c41-8c656aea4d6e%2Fycktglq_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given the equation:
\[ f(x) + x^2[f(x)]^3 = 10 \]
and
\[ f(1) = 2 \]
find \( f'(1) \).
### Detailed Explanation
We are provided with a functional equation involving \( f(x) \) and an initial condition for \( f \). Our goal is to find the first derivative of the function \( f \) at \( x = 1 \).
To solve this, we will:
1. Differentiate both sides of the given equation implicitly with respect to \( x \).
2. Substitute \( x = 1 \) and \( f(1) = 2 \) into the resulting equation.
3. Solve for \( f'(1) \).
### Solution Steps:
1. **Differentiate both sides of the equation implicitly:**
The given equation is:
\[ f(x) + x^2[f(x)]^3 = 10 \]
Taking the derivative with respect to \( x \), we apply the chain rule:
\[ \frac{d}{dx}\left[ f(x) \right] + \frac{d}{dx} \left[ x^2 (f(x))^3 \right] = \frac{d}{dx} [10] \]
Since the derivative of a constant is zero:
\[ f'(x) + \frac{d}{dx} \left[ x^2 (f(x))^3 \right] = 0 \]
2. **Differentiate the product \( x^2 (f(x))^3 \) using the product rule:**
Let \( u = x^2 \) and \( v = (f(x))^3 \). Then:
\[ \frac{d}{dx} [uv] = u'v + uv' \]
Here, \( u' = 2x \) and \( v = (f(x))^3 \):
\[ \frac{d}{dx} \left[ x^2 (f(x))^3 \right] = 2x (f(x))^3 + x^2 \left[ 3 (f(x))^2 f'(x) \right] \]
3. **Combine the derivatives:**
\[ f'(x) + 2x (f(x))^3 + x^2
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