Single Variable Calculus, Chapter 7, 7.6, Section 7.6, Problem 26

Determine the derivative of the function $y = \sqrt{x^2 - 1} \sec^{-1}(x)$ and simplify if possible.
If $y = \sqrt{x^2 - 1} \sec^{-1}(x)$, then by applying product rule and chain rule...

$
\begin{equation}
\begin{aligned}
y' &= (x^2 -1)^{\frac{1}{2}} \cdot \frac{d}{dx} \sec^{-1}(x) + \sec^{-1}(x) \cdot \frac{d}{dx} (x^2 -1)^{\frac{1}{2}}\\
\\
y' &= (x^2 -1)^{\frac{1}{2}} \cdot \left( \frac{1}{x\sqrt{x^2 - 1}} \right) + \sec^{-1}(x) \cdot \left[ \frac{1}{2}(x^2-1)^{-\frac{1}{2}} (2x) \right]\\
\\
y' &= \frac{1}{x} + \frac{x\sec^{-1}(x)}{\sqrt{x^2-1}}
\end{aligned}
\end{equation}
$