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The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
==Exponential functions==
Functions of the form <math>f(s)=a^{s}</math>, where ''a'' is constant and ''s'' is a [[complex number]] are known as exponential functions.
The derivative of exponential functions,
<math>\frac{d}{dx}a^{x}=s^{x}\ln{x}</math>
Where <math>\ln{s}</math> is the [[natural logarithm]] for ''x'' which is real.
The [[transcendental number]] ''[[e]]'' (2.7182818...) has the property that the [[derivative]] of the function ''e''<sup>''x''</sup> is ''e''<sup>''x''</sup>. This function is thus important in the solution of many types of differential equations.
The [[transcendental number]] ''[[e]]'' (2.7182818...) has the property that the [[derivative]] of the function ''e''<sup>''x''</sup> is ''e''<sup>''x''</sup>. This function is thus important in the solution of many types of differential equations.
[[Category:Mathematics]]
[[Category:Mathematics]]