Mathematics

Problem

Consider the relation

\sec y = x

, where

y

is an angle measured in radians and

x

satisfies

|x|>1

. This equation defines

y

implicitly as a function of

x

(that is,

y=\operatorname{arcsec}x

). (a) Differentiate both sides with respect to

x

to obtain an expression for

\dfrac{dy}{dx}

in terms of

y

and

x

. Show each step in your work. (b) Rewrite your result from part (a) so that

\dfrac{dy}{dx}

is expressed solely in terms of

x

and does not involve

y

. (c) Justify why an absolute value symbol appears in the simplified derivative and explain its significance in relation to the domain

|x|>1

. (d) Identify the open intervals on the real line where

\operatorname{arcsec}x

is differentiable and explain how your derivative formula supports this conclusion.

Intuitive explanation

The key is recognizing that $\sec y = x$ is the inverse of the secant function, so you're solving for the angle $y$ whose secant equals $x$ . Since $\sec y = \frac{1}{\cos y}$ , you're really asking: what angle has a cosine of $\frac{1}{x}$ ? The restriction $|x| > 1$ ensures $|\frac{1}{x}| < 1$ , which keeps the cosine value in its valid range. Understanding this reciprocal relationship and the domain constraint will guide you through any calculations involving $\operatorname{arcsec} x$ .

Intuitive explanation

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