True or False. If true, give an example. If false, explain why it is false and then provide the correct solution. If h(x) = f(x)g(x), then h' (x) = f '(x)g'(x). %3D Search entries or author Unread
True or False. If true, give an example. If false, explain why it is false and then provide the correct solution. If h(x) = f(x)g(x), then h' (x) = f '(x)g'(x). %3D Search entries or author Unread
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:**
Determine whether the following statement is true or false. Provide an example if the statement is true. If it is false, explain why and give the correct solution.
Statement:
If \( h(x) = f(x)g(x) \), then \( h'(x) = f'(x)g'(x) \).
**Solution Explanation:**
The statement is false. The correct method to find the derivative \( h'(x) \) when \( h(x) = f(x)g(x) \) is to use the product rule. The product rule states:
\[ h'(x) = f'(x)g(x) + f(x)g'(x) \]
**Explanation:**
- The derivative of the product of two functions is not simply the product of their derivatives. The product rule accounts for the variation in each function as well as the sum of the individual derivatives times the other function.
**Example:**
Let \( f(x) = x^2 \) and \( g(x) = x^3 \).
Then:
\[ h(x) = f(x)g(x) = x^2 \cdot x^3 = x^5 \]
Using the correct rule:
\[ h'(x) = 5x^4 \]
Using the incorrect statement:
If \( h'(x) = f'(x)g'(x) \), then:
\[ f'(x) = 2x \]
\[ g'(x) = 3x^2 \]
\[ h'(x) = 2x \cdot 3x^2 = 6x^3 \]
Clearly, \( 6x^3 \) is not equal to \( 5x^4 \). Hence, the correct application is using the product rule.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42185ff6-bd09-4fc4-949e-e31213ccb7f0%2F9931f8a3-fffb-4c2a-bf96-bff033ea00a5%2Fhqzn8v9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine whether the following statement is true or false. Provide an example if the statement is true. If it is false, explain why and give the correct solution.
Statement:
If \( h(x) = f(x)g(x) \), then \( h'(x) = f'(x)g'(x) \).
**Solution Explanation:**
The statement is false. The correct method to find the derivative \( h'(x) \) when \( h(x) = f(x)g(x) \) is to use the product rule. The product rule states:
\[ h'(x) = f'(x)g(x) + f(x)g'(x) \]
**Explanation:**
- The derivative of the product of two functions is not simply the product of their derivatives. The product rule accounts for the variation in each function as well as the sum of the individual derivatives times the other function.
**Example:**
Let \( f(x) = x^2 \) and \( g(x) = x^3 \).
Then:
\[ h(x) = f(x)g(x) = x^2 \cdot x^3 = x^5 \]
Using the correct rule:
\[ h'(x) = 5x^4 \]
Using the incorrect statement:
If \( h'(x) = f'(x)g'(x) \), then:
\[ f'(x) = 2x \]
\[ g'(x) = 3x^2 \]
\[ h'(x) = 2x \cdot 3x^2 = 6x^3 \]
Clearly, \( 6x^3 \) is not equal to \( 5x^4 \). Hence, the correct application is using the product rule.
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