<s>[INST] Differentiating secant inverse I want to show that $\cfrac{d}{dx}[\operatorname{arcsec} x]=\cfrac{1}{\vert x \vert\sqrt{x^2-1}}$. Here is my attempt:
\begin{align*} y=\sec^{-1}x &\iff x=\sec y \\ \frac{d}{dx}[x]&=\frac{d}{dx}[\sec y] \\ 1 &=\frac{\cos^2y}{\sin y} \frac{dy}{dx} \\ \frac{dy}{dx} &= \frac{\sin y}{\cos^2y} \end{align*}
I tried using the Pythagorean identity for 1 on the lefthand side, but couldn't see what else that would bring. I'm starting to think implicit differentiation isn't going to let me derive the expression I'm looking for.
Any ideas?
 [/INST] You have
$$x = \sec y = \frac{1}{\cos y} \implies \cos y = \frac{1}{x} \tag{1}\label{eq1A}$$
Also, you have, assuming that $\sin y \ge 0$, that
$$\sin^2 y + \cos^2 y = 1 \implies \sin y = \sqrt{1 - \cos^2 y} = \sqrt{1 - \frac{1}{x^2}} = \frac{\sqrt{x^2 - 1}}{|x|} \tag{2}\label{eq2A}$$
Note you have a mistake in your differentiation where you got the reciprocal of what you should have had. In particular,
$$\begin{equation}\begin{aligned}
\frac{d}{dx}[\sec y] & = \frac{d}{dx}\left(\frac{1}{\cos y}\right) \\
& = \left(-\frac{(-\sin y)}{\cos^2 y}\right)\frac{dy}{dx} \\
& = \left(\frac{\sin y}{\cos^2 y}\right)\frac{dy}{dx}
\end{aligned}\end{equation}\tag{3}\label{eq3A}$$
This then gives
$$\frac{dy}{dx} = \frac{\cos^2 y}{\sin y} \tag{4}\label{eq4A}$$
Using \eqref{eq1A} and \eqref{eq2A} in \eqref{eq4A} then gives the expression you're trying to prove, i.e, you get
$$\begin{equation}\begin{aligned}
\frac{dy}{dx} & = \frac{\frac{1}{x^2}}{\frac{\sqrt{x^2 - 1}}{|x|}} \\
& = \frac{|x|}{x^2\sqrt{x^2 - 1}} \\
& = \frac{1}{|x|\sqrt{x^2 - 1}}
\end{aligned}\end{equation}\tag{5}\label{eq5A}$$
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