Editing Exponential function (section)

===Differential equation===
[[Image:Exp tangent.svg|thumb|right |The derivative of the exponential function is equal to the value of the function. Since the derivative is the [[slope]] of the tangent, this implies that all green [[right triangle]]s have a base length of 1.]]

One of the simplest definitions is: The ''exponential function'' is the ''unique'' [[differentiable function]] that equals its [[derivative]], and takes the value {{math|1}} for the value {{math|0}} of its variable.

This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

''Uniqueness: ''If {{tmath|f(x)}} and {{tmath|g(x)}} are two functions satisfying the above definition, then the derivative of {{tmath|f/g}} is zero everywhere because of the [[quotient rule]]. It follows that {{tmath|f/g}} is constant; this constant is {{math|1}} since {{tmath|1=f(0) = g(0)=1}}.

''Existence'' is proved in each of the two following sections.