Derivative of Arcsecant Function/Corollary 1

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Corollary to Derivative of Arcsecant Function

Let $x \in \R$.

Let $\map \arcsec {\dfrac x a}$ be the arcsecant of $\dfrac x a$.


Then:

$\dfrac {\map \d {\map \arcsec {\frac x a} } } {\d x} = \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}$


Proof

\(\ds \frac {\map \d {\map \arcsec {\frac x a} } } {\d x}\) \(=\) \(\ds \frac 1 a \frac 1 {\size {\frac x a} \sqrt {\paren {\frac x a}^2 - 1} }\) Derivative of Arcsecant Function and Derivative of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac 1 a \frac 1 {\size {\frac x a} \frac {\sqrt {x^2 - a^2} } a}\)
\(\ds \) \(=\) \(\ds \frac 1 a \frac {a^2} {\size x {\sqrt {x^2 - a^2} } }\)
\(\ds \) \(=\) \(\ds \frac a {\size x {\sqrt {x^2 - a^2} } }\)

$\Box$


Similarly:

\(\ds \frac {\map \d {\map \arcsec {\frac x a} } } {\d x}\) \(=\) \(\ds \begin {cases} \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $\dfrac x a > 1$}) \\ \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $\dfrac x a < -1$}) \\ \end{cases}\) Derivative of Arcsecant Function
and Derivative of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end{cases}\) simplifying as above

$\blacksquare$


Also see