**Introduction**

This
article explains how to

*differentiate*, i.e. find the*derivative*of, the*arcsecant*function, which is seldom discussed in class. If you look at the various syllabus outlines, sometimes they do not explicitly mention or imply the arcsecant, neither do they explicitly exclude this.**The important thing is: we should be able to**The people who talk about and focus on mathematics syllabus content as if it is the only or most important thing are missing the point of mathematics education, and there are a plenty of these idiots around. Do not follow them.*figure it out*from our basic knowledge, which is the whole point of mathematics.
The arcsecant,
written as arcsec or sec

^{-1}, is also known as the*inverse secant*function. So*arcsecant*means that, given the value of a*secant*function, you want “the” angle whose secant is that given value. The problem is there are many possible values. Look at the graph below.**Defining the Inverse of the secant properly**

On
the graph, a horizontal line can pass through (infinitely) many points. Like any periodic trigonometric function, the
secant function is

*not*a*one-to-one*(a.k.a. “injective” or “one-one”) function. As such, it is*not invertible*. However, we can restrict the function so that its domain is [0,*p*] \ {*/*^{p}_{2}}. This is highlighted in yellow on the graph.
Remember
that sec

*x*=^{1}/_{cos x}. Basically we follow the principal values [0,*p*] of the*arccosine*function except*/*^{p}_{2}where the cosine is zero and its reciprocal the secant is undefined. With this restriction on the domain, we get a one-one secant function, with range (-¥, -1] È [1, +¥). We can now define the inverse function, and its graph is obtained by reflecting the above graph along the mirror line*y*=*x*. We get this:-
Observe
that in the yellow regions in both graphs, the gradients at the points are
non-negative. We have chosen the domain
of the secant function, which is the same as the range of the arcsecant
function, such that the derivatives will be non-negative.

**Deriving the Derivative**(refer to the“Onion” Method for differentiation)

**Remarks**

Do not
confuse arccos

*y*with (cos*y*)^{-1}. They are not mean the same thing.
In case you
are wondering, the prefix “arc-” means the angle, which (if you use

*radians*) is literally the same as the arc-length when the radius equals to 1. In symbols,*s*=*r**q*with*r*= 1 means*s*=*q*. Although the term can be used with*degrees*or other units, when doing advanced mathematics like calculus, we would usually be using radians anyway.**Suitable Levels**

*****GCE ‘A’ Levels H2 Mathematics

*****International Baccalaureate (IB) HL Mathematics

* Advanced Placement (AP) Calculus AB &
BC

* University / College calculus

* other syllabuses that involve differentiation

* any learner interested in calculus

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