We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of uses the interval , so that

Why would this be controversial? Let’s fast-forward a couple of semesters and use implicit differentiation (see also https://meangreenmath.com/2014/08/08/different-definitions-of-logarithm-part-8/ for how this same logic is used for other inverse functions) to find the derivative of :

At this point, the object is to convert the left-hand side to something involving only . Clearly, we can replace with . However, doing the same with is trickier. We begin with one of the Pythagorean identities:

So which is it, the positive answer or the negative answer? The answer is, without additional information, **we don’t know**!

- If (so that ), then is positive, and so .
- If (so that ), then is negative, and so .

We therefore have two formulas for the derivative of :

These may then be combined into the single formula

It gets better. Let’s now find the integral

Several calculus textbooks that I’ve seen will lazily give the answer

This answer works as long as , so that reduces to simply . For example, the red signed area in the above picture on the interval may be correctly computed as

However, the orange signed area on the interval is **incorrectly** computed using this formula!

This is patently false, as the picture clearly indicates that the above integral has to be negative. For this reason, careful calculus textbooks will often ask students to solve a problem like

and the caveat is needed to ensure that the correct antiderivative is used. Indeed, a calculus textbook that doesn’t include such caveats are worthy of any scorn that an instructor cares to heap upon it.