Find the derivative: frac{d}{dx} cos^{-1} lef | vedclass

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(C) Let $y = \cos^{-1} \left( \frac{x - \frac{1}{x}}{x + \frac{1}{x}} \right) = \cos^{-1} \left( \frac{x^2 - 1}{x^2 + 1} \right)$.Substitute $x = \cot

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$\frac{2}{1 + x^2}$

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(C) Let $y = \cos^{-1} \left( \frac{x - \frac{1}{x}}{x + \frac{1}{x}} ight) = \cos^{-1} \left( \frac{x^2 - 1}{x^2 + 1} ight)$.Substitute $x = \cot 	heta$,then $x^2 = \cot^2 	heta$.$y = \cos^{-1} \left( \frac{\cot^2 	heta - 1}{\cot^2 	heta + 1} ight) = \cos^{-1} \left( - \frac{1 - \cot^2 	heta}{1 + \cot^2 	heta} ight) = \cos^{-1} (- \cos 2	heta)$.Using the identity $\cos^{-1}(-z) = \pi - \cos^{-1}(z)$,we get $y = \pi - \cos^{-1}(\cos 2	heta) = \pi - 2	heta$.Since $x = \cot 	heta$,$	heta = \cot^{-1} x$.Thus,$y = \pi - 2 \cot^{-1} x$.Differentiating with respect to $x$:$\frac{dy}{dx} = 0 - 2 \left( - \frac{1}{1 + x^2} ight) = \frac{2}{1 + x^2}$.

Find the derivative: $\frac{d}{dx} \cos^{-1} \left( \frac{x - x^{-1}}{x + x^{-1}} \right)$

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