Derivative of Arcsecant Function/Corollary 2

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

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

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

Proof
From Derivative of Arcsecant Function:


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

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


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

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


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