e (mathematical Constant) - e in Calculus

e in 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 = ax has derivative given as the limit:

The limit on the right-hand side is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

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 derivative much simpler.

Another motivation comes from considering the base-a logarithm. Considering the definition of the derivative of log_a x as the limit:

where the substitution u = h/x was made in the last step. The last limit appearing in this calculation is again an undetermined limit that depends only on the base a, and if that base is e, the limit is one. So symbolically,

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

There are thus two ways in which to select a special number a = e. One way is to set the derivative of the exponential function ax to ax, 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. In fact, these two solutions for a are actually the same, the number e.