AP Calculus AB and BC glossary

Euler's number

Euler's number e is the constant approximately equal to 2.71828. It is the unique base for which the exponential function is its own derivative: the slope of e to the x at every point equals the height there. That self-matching property is why e, and its logarithm, are called natural.

The property that pins down $e$ is $\frac{d}{dx}\left[e^x\right] = e^x$ . It also shows up as a limit, $e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$ , and as the sum $\sum_{n=0}^{\infty}\frac{1}{n!}$ , both giving the same $2.71828\ldots$

For a general base, $\frac{d}{dx}\left[a^x\right] = a^x \ln a$ . That factor $\ln a$ equals 1 exactly when $a = e$ , which is the sense in which $e$ is the one base that differentiates cleanly. Any other exponential is really $e^{x \ln a}$ in disguise.

The mistake

Differentiating $e^x$ with the power rule to get $x\,e^{x-1}$ . The power rule needs the variable in the base; here the variable is in the exponent, and $\frac{d}{dx}\left[e^x\right] = e^x$ .

The clearest place $e$ shows up outside a maths class is compounding. Compound a balance more and more often, monthly, then daily, then every instant, and the growth factor converges on $e$ rather than running away: how continuous compounding works.

Appears in: Unit 2: Defining the Derivative