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Rudin to opine that the exponential function is "the most important function in mathematics".In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e.

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More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function.

This function property leads to exponential growth or exponential decay.

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The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.

One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 This is one of a number of characterizations of the exponential function; others involve series or differential equations.

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From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, .

The derivative (rate of change) of the exponential function is the exponential function itself.

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