Question

Find dy/dx if y= cos(sin(x)^1/2)^1/2

Answer 1

Question :-

$\sf \: find \: \frac{dy}{dx} \: if \: y = \cos {( \sin {x}^{ \frac{1}{2} } )}^{ \frac{1}{2} }$

Answer :-

$\boxed{ \to \sf \frac{dy}{dx} = \frac{ - \cos (\sqrt{x} )\sin\{ \sqrt{ \sin (\sqrt{x}) } \: \} }{4 \sqrt{x} \sqrt{ \sin( \sqrt{x} )} } \: } \:$

Solution :-

We have ,

$\sf \to \: y = \cos {( \sin {x}^{ \frac{1}{2} } )}^{ \frac{1}{2} } \\ \\ \sf we \: can \: write \: it \: \\ \\ \to \sf \: y = \cos \{ \sqrt{ \sin (\sqrt{x}) } \: \} \\ \\ \sf differentiate \: with \: respect \: to \: x \\ \\ \to \sf \: \frac{dy}{dx} = \frac{d}{dx} \cos \{ \sqrt{ \sin (\sqrt{x}) } \: \} \\ \\ \: \: \: \: \: \: \: \: \: \: \boxed{\because \sf \: \frac{d}{dx} \cos x = - \sin x} \\ \\ \to \sf \: \frac{dy}{dx} = - \sin \{ \sqrt{ \sin (\sqrt{x}) } \: \} \frac{d}{dx} \{ \sqrt{ \sin (\sqrt{x}) } \: \} \\ \\ \: \: \: \: \: \: \: \: \: \: \boxed{ \because \sf \frac{d}{dx} \: \sqrt{x} = \frac{1}{2 \sqrt{x} } } \\ \\ \to \sf \frac{dy}{dx} \: = \frac{ -\sin\{ \sqrt{ \sin (\sqrt{x}) } \: \} }{2 \sqrt{ \sin (\sqrt{x}) } } \: \frac{d}{dx} \sin(\sqrt{x} )\\ \\ \to \sf \frac{dy}{dx} = \frac{ -\sin\{ \sqrt{ \sin (\sqrt{x}) } \: \} }{2 \sqrt{ \sin (\sqrt{x}) } } \: \cos( \sqrt{x} ) \frac{d}{dx} \sqrt{x} \\ \\ \boxed{ \to \sf \frac{dy}{dx} = \frac{ - \cos (\sqrt{x} )\sin\{ \sqrt{ \sin (\sqrt{x}) } \: \} }{4 \sqrt{x} \sqrt{ \sin( \sqrt{x} )} } \: }$