Derivative of Arcsecant Function

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Theorem

Let $x \in \R$ be a real number such that $\size x > 1$.

Let $\arcsec x$ be the arcsecant of $x$.


Then:

$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$}) \\ \end{cases}$


Corollary 1

$\map {\dfrac \d {\d x} } {\map \arcsec {\dfrac x a} } = \dfrac a {\size x \sqrt {x^2 - a^2} } = \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $x > a$}) \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $x < -a$}) \\ \end{cases}$


Corollary 2

$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {x^2 \sqrt {1 - \frac 1 {x^2} } }$


Proof

Arcsecant Function


Let $y = \arcsec x$ where $\size x > 1$.

Then:

\(\ds y\) \(=\) \(\ds \arcsec x\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \sec y\) where $y \in \closedint 0 \pi \land y \ne \dfrac \pi 2$
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d y}\) \(=\) \(\ds \sec y \ \tan y\) Derivative of Secant Function
\(\ds \leadsto \ \ \) \(\ds \frac {\d y} {\d x}\) \(=\) \(\ds \dfrac 1 {\sec y \tan y}\) Derivative of Inverse Function
\(\ds \leadsto \ \ \) \(\ds \paren {\frac {\d y} {\d x} }^2\) \(=\) \(\ds \frac 1 {\sec^2 y \ \tan^2 y}\) squaring both sides
\(\ds \) \(=\) \(\ds \frac 1 {\sec^2 y \paren {\sec^2 y - 1} }\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds \frac 1 {x^2 \paren {x^2 - 1} }\) Definition of $x$
\(\ds \leadsto \ \ \) \(\ds \size {\dfrac {\d y} {\d x} }\) \(=\) \(\ds \dfrac 1 {\size x \sqrt {x^2 - 1} }\) squaring both sides


Since $\dfrac {\d y} {\d x} = \dfrac 1 {\sec y \tan y}$, the sign of $\dfrac {\d y} {\d x}$ is the same as the sign of $\sec y \tan y$.

Writing $\sec y \tan y$ as $\dfrac {\sin y} {\cos^2 y}$, it is evident that the sign of $\dfrac {\d y} {\d x}$ is the same as the sign of $\sin y$.


From Sine and Cosine are Periodic on Reals, $\sin y$ is never negative on its domain ($y \in \closedint 0 \pi \land y \ne \pi/2$).

However, by definition of the arcsecant of $x$:

$0 < \arcsec x < \dfrac \pi 2 \implies x > 1$
$\dfrac \pi 2 < \arcsec x < \pi \implies x < -1$

Thus:

$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin{cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$}) \\ \end{cases}$

Hence the result.

$\blacksquare$


Also see


Sources

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