Answer
${ \frac {(s|s|-1)}{|s|\sqrt{s^2-1}}}$ or ${ \frac {(s^2-1)}{s\sqrt{s^2-1}}}$
Work Step by Step
Given that $y$ = $\sqrt{{s^2-1}}-sec^{-1}{s}$, on applying the product rule of the derivative, we get:
$\frac{dy}{ds}$ = ${\frac{1}{2}\frac{}{\sqrt{s^2-1}}\frac{d({s^2-1})}{ds}-\frac{1}{|s|\sqrt{s^2-1}}}$
or $\frac{dy}{ds}$ = ${ \frac {2s}{2\sqrt{s^2-1}}-{\frac{1}{|s|\sqrt{(s^2-1)}}}}$
On simplifying:
$\frac{dy}{ds}$ = ${ \frac {(s|s|-1)}{|s|\sqrt{s^2-1}}}$ or ${ \frac {(s^2-1)}{s\sqrt{s^2-1}}}$
The final answer is: = ${ \frac {(s|s|-1)}{|s|\sqrt{s^2-1}}}$ or ${ \frac {(s^2-1)}{s\sqrt{s^2-1}}}$
