We’ve arrived at logarithms in our calculus class, and, of course, I explained that only holds for . That is all well and good.

And yes, I explained that expressions like only makes sense when

But then I went ahead and did a problem of the following type: given by using logarithmic differentiation,

And you KNOW exactly what I did. Right?

Note that is differentiable for all and, well, the derivative *should* be continuous for all but..is it? Well, up to inessential singularities, it is. You see: the second factor is not defined for , etc.

Well, let’s multiply it out and obtain:

So, there is that. We might induce inessential singularities.

And there is the following: in the process of finding the derivative to begin with we did:

and that expansion is valid only for

because we need and .

But the derivative formula works anyway. So what is the formula?

It is: if where is differentiable, then and verifying this is an easy exercise in induction.

But the logarithmic differentiation is really just a motivating idea that works for positive functions.

To make this complete: we’ll now tackle where it is essential that .

Rewrite

Then

This formula is a bit of a universal one. Let’s examine two special cases.

Suppose some constant. Then and the formula becomes which is just the usual constant power rule with the chain rule.

Now suppose for some positive constant. Then and the formula becomes which is the usual exponential function differentiation formula combined with the chain rule.