Question
Diffrentiate the following w.r.t.x
$\operatorname{cosec}(\sqrt{\cos X})$
$\operatorname{cosec}(\sqrt{\cos X})$
Differentiating w.r.t. x, we get
$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}[\operatorname{cosec}(\sqrt{\cos x})] \\ & =-\operatorname{cosec}(\sqrt{\cos x}) \cdot \cot (\sqrt{\cos x}) \cdot \frac{d}{d x} \sqrt{\cos x} \\ & =-\operatorname{cosec}(\sqrt{\cos x}) \cdot \cot (\sqrt{\cos x}) \cdot \frac{1}{2 \sqrt{\cos x}} \cdot \frac{d}{d x}(\cos x) \\ & =-\operatorname{cosec}(\sqrt{\cos x}) \cdot \cot (\sqrt{\cos x}) \cdot \frac{1}{2 \sqrt{\cos x}} \cdot(-\sin x) \\ & =\frac{\sin x \cdot \operatorname{cosec}(\sqrt{\cos x}) \cdot \cot (\sqrt{\cos x})}{2 \sqrt{\cos x}} .\end{aligned}$
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