College Math Placement Test 2025 – 400 Free Practice Questions to Pass the Exam

The function \( f(x) = e^x \) is unique in that its derivative is the same as the function itself. This property arises from the definition of the exponential function with base \( e \), which is approximately 2.71828. The derivative, which represents the rate of change of the function, tells us how steeply the function is increasing at any point \( x \).

To find the derivative of \( e^x \), we apply the fundamental rules of differentiation. The differentiation of the exponential function \( e^u \) with respect to \( x \) (where \( u = x \)) follows the chain rule. Since the derivative of \( u = x \) is 1, the derivative of \( e^x \) simplifies directly to \( e^x \).

Therefore, when taking the derivative, you find that \( f'(x) = e^x \). This result underlines one of the key characteristics of exponential functions, particularly those with base \( e \).

In contrast, the other choices involve incorrect forms and combinations of \( x \) and \( e \) that do not accurately reflect the derivative of \( e^x \). Thus, the correct answer—and indeed the accurate representation