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Suppose that 9(g(x))³ + 10x = 9x²g(x) + 40, and that g(4) = = −4. Find gʻ(4). g'(4) = ▶

Suppose that 9(g(x))³ + 10x = 9x²g(x) + 40, and that g(4) = = −4. Find gʻ(4). g'(4) = ▶

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:** Suppose that \( 9(g(x))^3 + 10x = 9x^2g(x) + 40 \), and that \( g(4) = -4 \). Find \( g'(4) \). **Solution:** To solve this problem, we need to differentiate both sides of the given equation with respect to \( x \) and then find \( g'(4) \) using the given condition. 1. Differentiate the equation \( 9(g(x))^3 + 10x = 9x^2g(x) + 40 \) with respect to \( x \). 2. Use the chain rule to differentiate \( (g(x))^3 \) and the product rule for \( x^2g(x) \). 3. Substitute \( g(4) = -4 \) into the derivative equation and solve for \( g'(4) \). Following these steps will yield the value of \( g'(4) \).
Transcribed Image Text:**Problem Statement:** Suppose that \( 9(g(x))^3 + 10x = 9x^2g(x) + 40 \), and that \( g(4) = -4 \). Find \( g'(4) \). **Solution:** To solve this problem, we need to differentiate both sides of the given equation with respect to \( x \) and then find \( g'(4) \) using the given condition. 1. Differentiate the equation \( 9(g(x))^3 + 10x = 9x^2g(x) + 40 \) with respect to \( x \). 2. Use the chain rule to differentiate \( (g(x))^3 \) and the product rule for \( x^2g(x) \). 3. Substitute \( g(4) = -4 \) into the derivative equation and solve for \( g'(4) \). Following these steps will yield the value of \( g'(4) \).
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