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Chapter 3: Problem 69

Determine whether the following statements are true and give an explanation or counterexample. a. \(\frac{d}{d x}\left(e^{5}\right)=5 e^{4}.\) b. The Quotient Rule must be used to evaluate $\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right).$ c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}.\) d. \(\frac{d}{d x}\left(x^{3} e^{x}\right)=3 x^{2} e^{x}.\)

Short Answer

Expert verified
Provide an explanation or a counterexample. Answer: The statement is false. The correct derivative is \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=-5x^{-6} = \frac{-5}{x^6}\). The given statement is incorrect since the correct derivative was found using the Power Rule on the expression \(x^{-5}\).

Step by step solution

01

Statement a

The statement is \(\frac{d}{d x}\left(e^{5}\right)=5 e^{4}\). To determine if this is true or false, we first need to take the derivative of \(e^5\) with respect to \(x\). Since \(e^5\) is a constant, its derivative with respect to \(x\) is simply zero. Therefore, the given statement is false. The correct derivative is: \(\frac{d}{d x}\left(e^{5}\right)=0\)
02

Statement b

The statement is that the Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\). It's true that we could use the Quotient Rule to evaluate this derivative, but it's not necessary, as we can simplify the given expression first. The given expression can be simplified as follows: \(\frac{x^{2}+3x+2}{x} = x + 3 + \frac{2}{x}\) Now, we can take the derivative of each term individually without needing the Quotient Rule: \(\frac{d}{d x}\left(x + 3 + \frac{2}{x}\right) = 1 - \frac{2}{x^2}\) So, the given statement is false, as the Quotient Rule is not necessary to find the derivative of the given expression.
03

Statement c

The statement is \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\). First, we can rewrite the given expression as \(x^{-5}\). Now, we can find the derivative using the Power Rule: \(\frac{d}{d x}\left(x^{-5}\right) = -5x^{-6}\) As we can see, the given statement is false. The correct derivative is: \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=-5x^{-6} = \frac{-5}{x^6}\)
04

Statement d

The statement is \(\frac{d}{d x}\left(x^{3} e^{x}\right)=3 x^{2} e^{x}\). To determine if this is true, we will need to use the Product Rule. The Product Rule states that the derivative of \(u(x)v(x)\) is: \(\frac{d}{d x}\left(u(x)v(x)\right) = u'(x)v(x) + u(x)v'(x)\) In this case, \(u(x) = x^3\) and \(v(x) = e^x\). Now, let's find the derivatives of \(u(x)\) and \(v(x)\): \(u'(x) = \frac{d}{d x}\left(x^{3}\right) = 3x^2\) \(v'(x) = \frac{d}{d x}\left(e^{x}\right) = e^x\) Applying the Product Rule: \(\frac{d}{d x}\left(x^{3} e^{x}\right) = 3x^2 e^x + x^3 e^x\) The given statement is false. The correct derivative is: \(\frac{d}{d x}\left(x^{3} e^{x}\right)=3 x^{2} e^{x} + x^3 e^x\)

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