Which expression is an antiderivative of e^x?

Which expression is an antiderivative of e^x?

• A. e^x + C
• B. e^{−x} + C
• C. e^{x^2} + C
• D. x e^x + C

Answer: (A)

An antiderivative is a function whose derivative returns the given function. For f(x) = e^x, the function e^x is its own derivative, so its antiderivatives are e^x + C, since differentiating e^x + C gives e^x. Differentiating the other expressions does not yield e^x: derivative of e^{-x} is -e^{-x}, not e^x; derivative of e^{x^2} is 2x e^{x^2}; and derivative of x e^x is (1+x) e^x. Only e^x + C has the correct derivative, capturing all antiderivatives with the constant of integration.

An antiderivative is a function whose derivative returns the given function. For f(x) = e^x, the function e^x is its own derivative, so its antiderivatives are e^x + C, since differentiating e^x + C gives e^x.

Differentiating the other expressions does not yield e^x: derivative of e^{-x} is -e^{-x}, not e^x; derivative of e^{x^2} is 2x e^{x^2}; and derivative of x e^x is (1+x) e^x. Only e^x + C has the correct derivative, capturing all antiderivatives with the constant of integration.