Step 1: Understanding the Concept:
The question asks for a function \( f(x) \) such that \( \frac{d}{dx}[f(x)] = f(x) \). This is a unique property of the exponential function with base \( e \). Step 2: Detailed Explanation:
Let's check the derivatives for all options:
- (A) \( \frac{d}{dx}(\sin x) = \cos x \neq \sin x \)
- (B) \( \frac{d}{dx}(\cos x) = -\sin x \neq \cos x \)
- (C) \( \frac{d}{dx}(\log x) = \frac{1}{x} \neq \log x \)
- (D) \( \frac{d}{dx}(e^{x}) = e^{x} \cdot \frac{d}{dx}(x) = e^{x} \).
Here, the function and its derivative are identical. Step 3: Final Answer:
The function is \( e^{x} \).