where k is a constant. Everyday examples for biology include bacterial growth and radioactive decay. As most of you know, we can solve this problem using a fundamental result from calculus:

So, rearranging the first equation and then integrating:

Evaluate the constant A by setting t = 0, so then x = x

_{0}:

At a special value, t = T (one generation, one half-life):

I remember reading something by Francois Jacob, a long time ago, in which he described his and Jacques Monod's excitement at some point that they had done a set of experiments and the value 0.69 appeared from their calculations for all the experiments. What was this magic number? It took a minute before the penny dropped, and they remembered that ln(2) = 0.693:

I haven't been able to find the reference.

After that long preamble, here is the real subject of this post. Starting from:

we want to show that:

By definition:

Take the derivative on both sides:

The right side equals 1. That is:

Now for the left side. It is easier if we make a substitution. Let:

And

We use the chain rule:

By the definition of e above in (1):

So:

Here is the trick. Substitute u = ln(x) to go back to x:

But from (2), the left side equals 1:

Or, quoting Heath's translation of Euclid: "precisely what was required to be proved."

I would love to credit the source for this, but I've forgotten where I found it.