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Chapter 2: Problem 75

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f(x)=x^{2 n}\), where \(n\) is an integer, then $f^{\prime}(x)=2 n x^{2(n-1)}$.

Short Answer

Expert verified
The statement is false. When differentiating the given function \(f(x) = x^{2n}\) using the Power Rule, we obtain the derivative \(f^{\prime}(x) = (2n)x^{(2n-1)}\), which is not equivalent to the given derivative \(f^{\prime}(x) = 2nx^{2(n-1)}\). A counterexample to the statement is when \(n=1\), the correct derivative for \(f(x) = x^2\) is \(f^{\prime}(x) = 2x\), not the given \(f^{\prime}(x) = 2\).

Step by step solution

01

Identify the given function and its derivative

The given function is \(f(x) = x^{2n}\), where \(n\) is an integer. The given derivative is \(f^{\prime}(x) = 2nx^{2(n-1)}\).
02

Differentiate the function using the Power Rule

To differentiate the function \(f(x) = x^{2n}\), we will use the Power Rule, which states that if \(f(x) = x^m\) for some real number \(m\), then \(f^{\prime}(x) = mx^{m-1}\). Applying the Power Rule to our function, we get: \(f^{\prime}(x) = (2n)x^{(2n-1)}\)
03

Compare the calculated derivative to the given derivative

Now, let's compare our calculated derivative, \((2n)x^{(2n-1)}\), to the given derivative, \(2nx^{2(n-1)}\). Are these expressions equivalent? To determine if they are equivalent, we need to see if the exponents are the same: \(2n - 1 = 2(n - 1)\) Expanding the right side of the equation, we get: \(2n - 1 = 2n - 2\)
04

Determine the truth of the statement and provide an explanation

When comparing the exponents, we find that the equation \(2n - 1 = 2n - 2\) is false, so the given statement is false. To show why the statement is false, let's look at an example: Let \(n = 1\). Then, we have: \(f(x) = x^{2(1)} = x^2\) The given derivative is: \(f^{\prime}(x) = 2(1)x^{2(1-1)} = 2x^0 = 2\) However, the correct derivative for \(f(x) = x^2\) is: \(f^{\prime}(x) = 2x\) Thus, we have shown that the statement is false by providing an example with \(n=1\).

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