Question

What is the derivative of ${\sec }^{-1}\left(x\right)$?

Answer 1

Find derivative of ${\sec }^{-1}\left(x\right)$

Let $y={\sec }^{-1}\left(x\right)$

Differentiate it w.r.t $x$

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}{\sec }^{-1}\left(x\right) \\ =\frac{1}{\left|x\right|\sqrt{x^2-1}},x\ne \pm 1,0 \end{gathered}$$

Hence, derivative of ${\sec }^{-1}\left(x\right)$ is $\frac{1}{\left|x\right|\sqrt{x^2-1}},x\ne \pm 1,0$.