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The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own [[Derivative|derivatives]]:
: <math>{d \over dx} a^x = (\ln
If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population growth, continuously compounded interest or radioactive decay, then the variable can be written as a constant times an exponential function of time.
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The exponential function thus solves the basic [[differential equation]]
: <math>{dy \over dx} = y</math>
and it is for this reason commonly encountered in differential equations. In particular the solution of linear ordinary [[differential equation]]s can frequently be written in terms of exponential functions. These equations include [[Schrödinger equation]] and the [[Laplace's equation]] as well as the equations for [[simple harmonic motion]].
=== Exponential function on the complex plane ===
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