Derivative of Arcsecant Function/Corollary 2

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Theorem

Let $x \in \R$ be a real number such that $x < -1$ or $x > 1$.

Let $\operatorname{arcsec} x$ be the arcsecant of $x$.


Then:

$\dfrac {\mathrm d \left({\operatorname{arcsec} x }\right)} {\mathrm d x} = \dfrac 1 {x^2 \sqrt {1 - \frac 1 {x^2}}}$


Proof

From Derivative of Arcsecant Function:

$\dfrac {\mathrm d \left({\operatorname{arcsec} x}\right)} {\mathrm d x} = \dfrac 1 {\left|{x}\right| \sqrt {x^2 - 1} }$


Since for all $x \in \R$, we have $\left|{x}\right| = \sqrt{x^2}$, we can write:

$\dfrac {\mathrm d \left({\operatorname{arcsec} x}\right)} {\mathrm d x} = \dfrac 1 {\sqrt {x^2} \sqrt {x^2 - 1}}$

Multiplying the denominator by $1 = \dfrac {\sqrt{x^2}} {\sqrt{x^2}}$ yields:

$\dfrac {\mathrm d \left({\operatorname{arcsec} x}\right)} {\mathrm d x} = \dfrac 1 {x^2 \sqrt {1 - \frac 1 {x^2}}}$

$\blacksquare$


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