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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Question
![**Problem Statement:**
Let
\[ f(x) = \sqrt[4]{6x^2 + 9x + 4}. \]
Use the chain rule to find
\[ \frac{d}{dx} f(x) \bigg|_{x=1}. \]
\[ \frac{d}{dx} f(x) \bigg|_{x=1} = \_ . \]
---
**Instructions:**
To solve this problem, apply the chain rule to find the derivative of \( f(x) \) and compute it at \( x = 1 \). Follow these steps:
1. **Identify Inner and Outer Functions:**
- Consider the inner function \( g(x) = 6x^2 + 9x + 4 \).
- The outer function is \( h(u) = \sqrt[4]{u} \), where \( u = g(x) \).
2. **Differentiate Inner and Outer Functions:**
- Find the derivative of the inner function \( g(x) \).
- Find the derivative of the outer function \( h(u) \).
3. **Apply the Chain Rule:**
- Use the chain rule: \[ \frac{d}{dx} f(x) = h'(g(x)) \cdot g'(x) \]
4. **Evaluate at \( x = 1 \):**
- Substitute \( x = 1 \) into the derivative and solve for the specific value.
Fill in the blank with the final derivative evaluation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1467f162-7185-45ab-925a-2589d3c8cce7%2Fbe6649ee-d86a-4fcd-9e46-06c68f519182%2Fg5m5msp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let
\[ f(x) = \sqrt[4]{6x^2 + 9x + 4}. \]
Use the chain rule to find
\[ \frac{d}{dx} f(x) \bigg|_{x=1}. \]
\[ \frac{d}{dx} f(x) \bigg|_{x=1} = \_ . \]
---
**Instructions:**
To solve this problem, apply the chain rule to find the derivative of \( f(x) \) and compute it at \( x = 1 \). Follow these steps:
1. **Identify Inner and Outer Functions:**
- Consider the inner function \( g(x) = 6x^2 + 9x + 4 \).
- The outer function is \( h(u) = \sqrt[4]{u} \), where \( u = g(x) \).
2. **Differentiate Inner and Outer Functions:**
- Find the derivative of the inner function \( g(x) \).
- Find the derivative of the outer function \( h(u) \).
3. **Apply the Chain Rule:**
- Use the chain rule: \[ \frac{d}{dx} f(x) = h'(g(x)) \cdot g'(x) \]
4. **Evaluate at \( x = 1 \):**
- Substitute \( x = 1 \) into the derivative and solve for the specific value.
Fill in the blank with the final derivative evaluation.
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