Exponential function

exponential functions

Exponential functions are functions of the form $f(s)=a^{s}$, where a is a constant and s is a complex number. They are based on the mathematical operation of exponentiation.

The derivative of an exponential function is,

${\frac {d}{dx}}a^{x}=a^{x}\ln {a}$,

where $\ln {a}$ is the natural logarithm for a.

The transcendental number e (2.7182818...) has the property that the derivative of the function e^x is e^x, since ln e = 1. This function is thus important in the solution of many types of differential equations. e is the canonical exponential function.