Derivative of root cosec x

Let y = \sqrt{cosec x} \Rightarrow y = {(cosec x)}^{1 / 2} \Rightarrow \log y = \log {(cosec x)}^{1 / 2} \Rightarrow \log y = \frac{1}{2} \log (cosec x) differentiating both sides with respect to x, we get \frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \times \frac{1}{cosec x} \times - \cot x . cosec x \Rightarrow \frac{1}{y} \frac{dy}{dx} = - \frac{\cot x}{2} \Rightarrow \frac{dy}{dx} = - y \times \frac{\cot x}{2} \Rightarrow \frac{dy}{dx} = - \frac{1}{2} \sqrt{cosec x} . \cot x

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