## Wednesday, August 5, 2009

The exponential function comes up all the time in science. Often the rate of change of some quantity x is proportional to the value x itself:

 `dx/dt = k * x`

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:

 `d/dx ln(x) = 1/xln(x) = ∫ 1/x dx`

So, rearranging the first equation and then integrating:

 `dx/x = k * dt∫ dx/x = ∫ k * dtln(x) = k * tx = A exp { k * t }`

Evaluate the constant A by setting t = 0, so then x = x0:

 `x = x0 exp { k * t }`

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

 `x = 2 * x02 = exp { k * t }ln(2) = k * T`

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:

 `>>> import math>>> print round(log(2),3)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:

 `(1) d/dx ex = ex`

we want to show that:

 `d/dx ln(x) = 1/x`

By definition:

 ` eln(x) = x`

Take the derivative on both sides:

 ` d/dx eln(x) = d/dx x`

The right side equals 1. That is:

 `(2) d/dx eln(x) = 1`

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

 ` u = ln(x) eln(x) = eu = x`

And

 ` d/dx eln(x) = d/dx eu`

We use the chain rule:

 ` d/dx eu = d/du eu * du/dx`

By the definition of e above in (1):

 ` d/du eu = eu`

So:

 ` d/dx eu = eu * du/dx`

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

 ` d/dx eu = x * du/dx d/dx eln(x) = x * du/dx d/dx eln(x) = x * d/dx ln(x)`

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

 ` 1 = x * d/dx ln(x) d/dx ln(x) = 1/x`

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.