Answer
$- sech^{-1} x$
Work Step by Step
Given: $y=\cos^{-1} x-x sech^{-1}x$
Since, $\dfrac{d (\cos^{-1} x)}{dx}=-\dfrac{1}{\sqrt{1- x^2}}$ and $\dfrac{d (sech^{-1} x)}{dx}=\dfrac{-1}{x \sqrt{1- x^2}}$
Apply product rule to get the derivative:
Thus, $\dfrac{dy}{dx}=[-\dfrac{1}{\sqrt{1- x^2}}] [ sech^{-1} x (1) +(x) \dfrac{-1}{x \sqrt{1- x^2}}]$
or, $=-\dfrac{1}{\sqrt{1- x^2}}- sech^{-1} x+ \dfrac{1}{ \sqrt{1- x^2}}$
Hence, $\dfrac{dy}{dx}=- sech^{-1} x$
