The derivative of e^x is:

• A. 1/e^x
• B. e^{x+1}
• C. x e^x
• D. e^x

Answer: (D)

Exponential functions with base e change at a rate equal to their value. In other words, the derivative of e^x with respect to x is e^x. This comes from the general rule d/dx a^x = ln(a) · a^x; since ln(e) = 1, you get d/dx e^x = e^x.

Seeing the other forms helps: d/dx e^{x+1} = e^{x+1} (which is e·e^x, not e^x), d/dx e^{-x} = -e^{-x}, and d/dx (x e^x) = e^x + x e^x (not just e^x). So the correct derivative is e^x.