Content deleted Content added
many small edits; rearranged & rewrote some bits; right-aligned graph again (why not?); see Talk for more info |
disambiguating "proportionality" |
||
Line 40:
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their [[derivative]]s. In particular,
: <math>{d \over dx} e^x = e^x</math>
That is, ''e''<sup>''x''</sup> is its own [[derivative]], a property unique among real-valued functions of a real variable. Other ways of saying the same thing include:
*The slope of the graph at any point is the height of the function at that point.
Line 49 ⟶ 51:
For exponential functions with other bases:
: <math>{d \over dx} a^x = (\ln a) a^x</math>
Thus ''any'' exponential function is a [[constant]] multiple of its own derivative.
If a variable's growth or decay rate is [[proportionality (mathematics)|proportional]] to its size — as is the case in unlimited population growth (see [[Malthusian catastrophe]]), continuously compounded [[interest]], or [[radioactive decay]] — then the variable can be written as a constant times an exponential function of time.
==Formal definition==
| |||