Then our news item can be written

This formula can be rewritten in terms of the

where it is now to be

This means that the dollar in your pocket right now will be worth only $0.99999996829 in one second. Well, this is interesting, but we cannot go any further with it until we ask a crucial question: "What will happen if this goes on?" That is, suppose we assume that equation (2) is not just a temporary situation, but

Applying the *d*/*dt* "operator" to both sides
of Eq. (2) gives

But

That is, the rate of change of the rate of change is always positive, or the (negative) rate of change is getting

So by noting the initial value of *V*, which is formally
written *V*_{0} but in this case equals $1.00, and by applying
our understanding of the "graphical meaning"
of the first derivative (slope) and the second derivative
(curvature), we can visualize the function *V*(*t*) pretty well.
It starts out with a maximum downward slope
and then starts to level off as time increases.
This general trend continues indefinitely.
Note that while the function always decreases,
it *never reaches zero*.
This is because, the closer it gets to zero,
the slower it decreases [see Eq. (2)].
This is a very "cute" feature that makes this function
especially fun to imagine over long times.

We can also apply our analytical understanding
to the formulas (2) and (4) for the derivatives:
every time we take still another derivative, the result
is still proportional to *V* - the constant of proportionality
just picks up another factor of (- 0.1). This is
a *really neat* feature of this function, namely
that we can write down *all its derivatives*
with almost no effort:

This is a pretty nifty function. What *is* it?
That is, can we write it down in terms of
familiar things like *t*, *t*^{2}, *t*^{3}, and so on?
First, note that Eq. (9) can be written in the form

A simpler version would be where

follows by simple "differentiation" [a single word for "taking the derivative"]. Now, these two equations have similar-looking right-hand sides, provided that we pretend not to notice that term in

(12) |

Now, suppose we give

(read, "

(14) |

We could also write a more abstract version of this function in terms of a generalized variable "

Let's do this, and then define and set . Then, by the CHAIN RULE for derivatives,

(16) |

and since , we have

(17) |

By repeating this we obtain Eq. (10). Thus

where

(19) |

In FORTRAN it is represented as

(20) |

raised to the power. In our case we have , so that our "answer" is

which is a lot easier to write down than Eq. (18). Now, the choice of notation

(22) |

You can easily determine that this rule also works for the definition in Eq. (15).

The "inverse" of this function (the power to which
one must raise *e* to obtain a specified number)
is called the "**natural logarithm**" or "" function.
We write

or

(23) |

A handy application of this definition is the rule

(24) |

Before we return to our original function, is there
anything more interesting about the "natural logarithm"
than that it is the inverse of the "exponential" function?
And what is so all-fired special about *e*, the "base"
of the natural log? Well, it can easily be shown^{9}
that the *derivative* of
is a very simple and familiar function:

(25) |

This is perhaps the most useful feature of , because it gives us a direct connection between the exponential function and a function whose derivative is 1/

Jess H. Brewer - Last modified: Fri Nov 13 17:21:02 PST 2015