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Calculus

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential function y = a has a derivative, given by a limit:

The parenthesized limit on the right is independent of the variable x. Its value turns out to be the logarithm of a to base e. Thus, when the value of a is set to e, this limit is equal to 1, and so one arrives at the following simple identity:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler.

Another motivation comes from considering the derivative of the base- a logarithm (i.e., log x), for x > 0:

where the substitution u = h / x was made. The base- a logarithm of e is 1, if a equals e. So symbolically,

The logarithm with this special base is called the natural logarithm, and is usually denoted as ln; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

Thus, there are two ways of selecting such special numbers a. One way is to set the derivative of the exponential function a equal to a, and solve for a. The other way is to set the derivative of the base a logarithm to 1/ x and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for a are actually the same: the number e.

The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0: e x = ∑ n = 0 ∞ x n n !. {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.} Setting x = 1 {\displaystyle x=1} recovers the definition of e as the sum of an infinite series.

The natural logarithm function can be defined as the integral from 1 to x {\displaystyle x} of 1 / t {\displaystyle 1/t}, and the exponential function can then be defined as the inverse function of the natural logarithm. The number e is the value of the exponential function evaluated at x = 1 {\displaystyle x=1}, or equivalently, the number whose natural logarithm is 1. It follows that e is the unique positive real number such that ∫ 1 e 1 t d t = 1. {\displaystyle \int _{1}^{e}{\frac {1}{t}}\,dt=1.}

Because e is the unique function (up to multiplication by a constant K) that is equal to its own derivative,

d d x K e x = K e x, {\displaystyle {\frac {d}{dx}}Ke^{x}=Ke^{x},}

it is therefore its own antiderivative as well:

∫ K e x d x = K e x + C. {\displaystyle \int Ke^{x}\,dx=Ke^{x}+C.}

Equivalently, the family of functions

y (x) = K e x {\displaystyle y(x)=Ke^{x}}

where K is any real or complex number, is the full solution to the differential equation

y ′ = y. {\displaystyle y'=y.}