Derivative of Inverse CSC

Problem 158: Use the Identity

(1) $\begin{equation*} \csc^{-1} u = \frac{\pi}{2} - \sec^{-1}u\end{equation*}$

to derive the formula for the derivative of $\csc^{-1} u$ .

Solution: Using the identity (1), we have

(2) $\begin{align*} \csc^{-1} u & = \frac{\pi}{2} - \sec^{-1}u \\ \frac{d}{du} \csc^{-1} u & = \frac{d}{du} \left( \frac{\pi}{2} - \sec^{-1} u \right) \\ & = - \frac{d}{du} \sec^{-1} u \\ & = - \frac{1}{|u| \sqrt{u^2-1}} , \ |u| > 1. \end{align*}$

That is,

(3) $\begin{equation*} \frac{d}{du} \csc^{-1} u = - \frac{1}{|u|\sqrt{u^2-1}}, \ |u| > 1. \end{equation*}$

Leave a Reply Cancel reply