AP Calculus AB and BC glossary

Natural exponential function

Also called: Natural exponential, Exponential function with base e

The natural exponential function is e to the x, where e is about 2.71828. It is the unique exponential function that is its own derivative, so at every point its slope equals its height. Its antiderivative is itself plus a constant.

\frac{d}{dx}e^{x} = e^{x}, \quad \int e^{x}\,dx = e^{x} + C

Every exponential is its own derivative up to a constant factor, since $\frac{d}{dx}a^{x} = a^{x}\ln a$ . That factor is $\ln a$ , and it equals 1 for exactly one base. The base is $e \approx 2.71828$ , so $e$ is not a convenient constant that happens to work. It is defined by the requirement.

That is also why any exponential can be rewritten in base $e$ . Since $a = e^{\ln a}$ , you can replace $a^{x}$ with $e^{x\ln a}$ and differentiate with the chain rule alone. Composites follow the same pattern: $\frac{d}{dx}e^{u} = e^{u}\cdot\frac{du}{dx}$ , so $e^{-x^{2}}$ differentiates to $-2xe^{-x^{2}}$ .

The mistake

Assuming anything with an $e$ in it reproduces itself. Only $e^{x}$ and its constant multiples are unchanged; $Ce^{x}$ is the complete set of functions equal to their own derivative. $e^{3x}$ differentiates to $3e^{3x}$ and integrates to $\frac{1}{3}e^{3x} + C$ , and $e^{x^{2}}$ has no elementary antiderivative at all, so no substitution will ever crack it.

Appears in: Unit 2: Defining the Derivative, Unit 3: Chain Rule, Implicit, and Inverses, Unit 6: Integration and Accumulation