Here is one way to think about this problem:

You are given a function, , defined by:

(This is shown in the attached graph, below.)

Now, instead of tying up one quantity as a function of the other, we can view this as a relationship between the quantities and , which is split into three regions:

- In the region where (equivalently ), they are related by ;
- In the region (equivalently ), they are related by .
- In the region , .
(A little technical point: I've used a lot of and signs in this definition - this is mostly okay as the function is; At , both expressions for give the same value.)continuous

By looking at the problem in this way, we can try to describe the relationship with aninversefunction, :

(Another small technical point: is not defined for (it takes infinitely many values here!), or for as is never negative.)

- - -

That was a bit of excess detail, but it pays off now. You are asked to consider a quantity called :

A big question is: what is a function of? ? ? Both?

The answer is, we can make it whichever we want! By expressing in terms of , we can make only; but by expressing in terms of , we can make only.

Let's do the latter, and make :

Now, evaluate the derivative :

as required.