91Ó°ÊÓ

Let's consider the function \(f(x) = |x|\). To find its domain, we need to determine the set of all valid input values for the function. The function \(f(x) = |x|\) is defined for all real numbers, meaning the domain is the entire real number line, \(\mathbb{R}\). Now let's find the derivative of \(f(x)\) to determine its domain: The derivative of \(|x|\) is defined piecewise as follows: \[f'(x) = \begin{cases} 1, & \text{if } x > 0 \\ -1, & \text{if } x < 0 \\ \text{undefined}, & \text{if } x = 0 \end{cases}\] We see that the derivative, \(f'(x)\), is undefined at \(x = 0\). Thus, the domain of the derivative function is \(\mathbb{R}\) except for \(x = 0\). This example demonstrates that the domain of \(f'\) is not the same as that of \(f\) for this specific example.

The statement "The domain of \(f^{\prime}\) is the same as that of \(f\)" is false. We can see this by analyzing the function \(f(x) = |x|\) where the domain of the original function, \(f\), is the entire real number line, but the derivative function, \(f'(x)\), has a domain that excludes \(x = 0\).