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Prove that (d)/(dx)("cosec"^(-1)x)=(-1)/...

Prove that `(d)/(dx)("cosec"^(-1)x)=(-1)/(|x|sqrt(x^(2)-1))`, where `x in R-[-1,1]`.

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To prove that \(\frac{d}{dx}(\csc^{-1}x) = \frac{-1}{|x|\sqrt{x^2 - 1}}\) for \(x \in \mathbb{R} - [-1, 1]\), we can follow these steps: ### Step 1: Define the function Let \(y = \csc^{-1}(x)\). This implies that \(x = \csc(y)\). ### Step 2: Differentiate using implicit differentiation To find \(\frac{dy}{dx}\), we differentiate both sides with respect to \(x\): \[ ...
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