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 = x0:
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:
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:
We use the chain rule:
By the definition of e above in (1):
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.