Definition of Exponential Functions

Exponential functions are generally given in the form . They are continuous for all values of x, and have inverses that are logarithmic functions. The derivative of an exponential function is , which is based on the chain rule and a restatement of a as (because the natural exponent and the natural log are inverse functions of one another). The power rule does not apply to exponential functions, because the variable is the exponent rather than the base. In the special case of the natural exponent (also known as Euler's number), the first derivative is . Euler's number e is special because it is irrational and not a root of any non-zero polynomial that has rational coefficients.