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146. Think About It Let f and g be functions whose first and second derivatives exist on an interval I. Which of the following formulas is (are) true? (a) fg" – f"g = (fg' – f'g)' (b) fg" + f"g = (fg)" - %D

146. Think About It Let f and g be functions whose first and second derivatives exist on an interval I. Which of the following formulas is (are) true? (a) fg" – f"g = (fg' – f'g)' (b) fg" + f"g = (fg)" - %D

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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Please solve 146 

### Educational Content on Derivatives and Product Rule #### Misconceptions About Derivatives When studying calculus, it's important to understand and correct some common misconceptions regarding derivatives. Here’s a list of statements. Some are true, others are false, and we will discuss examples or counterexamples accordingly. 139. **Statement**: If \( y = f(x)g(x) \), then \(\frac{dy}{dx} = f'(x)g'(x)\). 140. **Statement**: If \( y = (x+1)(x+2)(x+3)(x+4) \), then \(\frac{d^5y}{dx^5} = 0\). 141. **Statement**: If \( f'(c) \) and \( g'(c) \) are zero and \( h(x) = f(x)g(x) \), then \( h'(c) = 0\). 142. **Statement**: If the position function of an object is linear, then its acceleration is zero. 143. **Statement**: The second derivative represents the rate of change of the first derivative. 144. **Statement**: The function \( f(x) = \sin x + c \) satisfies \( f^{(n)} = f^{(n+4)} \) for all integers \( n \ge 1 \). #### Application of the Product Rule 145. **Proof**: Use the Product Rule twice to prove that if \( f, g, \) and \( h \) are differentiable functions of \( x \), then \[ \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x). \] #### Exploring Second Derivatives 146. **Think About It**: Let \( f \) and \( g \) be functions whose first and second derivatives exist on an interval \( I \). Which of the following formulas is (are) true? (a) \( fg'' - f''g = (f'g' - f'g)' \) (b) \( fg'' + f''g = (fg)'' \) In exploring these concepts, detailed analysis and application of derivative rules such as the product rule, chain rule, and higher-order derivatives are essential.
Transcribed Image Text:### Educational Content on Derivatives and Product Rule #### Misconceptions About Derivatives When studying calculus, it's important to understand and correct some common misconceptions regarding derivatives. Here’s a list of statements. Some are true, others are false, and we will discuss examples or counterexamples accordingly. 139. **Statement**: If \( y = f(x)g(x) \), then \(\frac{dy}{dx} = f'(x)g'(x)\). 140. **Statement**: If \( y = (x+1)(x+2)(x+3)(x+4) \), then \(\frac{d^5y}{dx^5} = 0\). 141. **Statement**: If \( f'(c) \) and \( g'(c) \) are zero and \( h(x) = f(x)g(x) \), then \( h'(c) = 0\). 142. **Statement**: If the position function of an object is linear, then its acceleration is zero. 143. **Statement**: The second derivative represents the rate of change of the first derivative. 144. **Statement**: The function \( f(x) = \sin x + c \) satisfies \( f^{(n)} = f^{(n+4)} \) for all integers \( n \ge 1 \). #### Application of the Product Rule 145. **Proof**: Use the Product Rule twice to prove that if \( f, g, \) and \( h \) are differentiable functions of \( x \), then \[ \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x). \] #### Exploring Second Derivatives 146. **Think About It**: Let \( f \) and \( g \) be functions whose first and second derivatives exist on an interval \( I \). Which of the following formulas is (are) true? (a) \( fg'' - f''g = (f'g' - f'g)' \) (b) \( fg'' + f''g = (fg)'' \) In exploring these concepts, detailed analysis and application of derivative rules such as the product rule, chain rule, and higher-order derivatives are essential.
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