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Let Use the chain rule to find f(x)|x=1 f(x) = √6x² + 9x + 4. d - f(x)|x=1· dx

Let Use the chain rule to find f(x)|x=1 f(x) = √6x² + 9x + 4. d - f(x)|x=1· dx

Calculus: Early Transcendentals
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:** 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.
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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