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Question Number 187695 by mnjuly1970 last updated on 20/Feb/23

$$$$$$If,f\left(x\right)=\frac{si\underset{}{n}\left(cos\left(x\right)\right)}{\stackrel{}{}\sqrt{\frac{\pi }{x}}}$$$$\Rightarrow f{}^{\prime }\left(\frac{\pi }{2}\right)=?$$

Answered by horsebrand11 last updated on 20/Feb/23

$$f\left(x\right)=\frac{\sqrt{x}\sin \left(\cos x\right)}{\sqrt{\pi }}$$$$f{}^{\prime }\left(\frac{\pi }{2}\right)=\underset{x\to \frac{\pi }{2}}{\lim }\frac{f\left(x\right)-f\left(\frac{\pi }{2}\right)}{x-\frac{\pi }{2}}$$$$=\frac{1}{\sqrt{\pi }}\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sqrt{x}\sin \left(\cos x\right)-\sqrt{\frac{\pi }{2}}\sin \left(\cos \frac{\pi }{2}\right)}{x-\frac{\pi }{2}}$$$$=\frac{1}{\sqrt{\pi }}.\frac{\sqrt{\pi }}{\sqrt{2}}.\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(\cos x\right)}{x-\frac{\pi }{2}}$$$$=\frac{1}{\sqrt{2}}.\underset{x\to 0}{\lim }\frac{\sin (-\sin x)}{x}$$$$=\frac{1}{\sqrt{2}}.\underset{x\to 0}{\lim }\frac{-\sin x}{x}=-\frac{1}{\sqrt{2}}$$

Answered by cortano12 last updated on 20/Feb/23

$$f\left(x\right)=\frac{\sqrt{x}\sin \left(\cos x\right)}{\sqrt{\pi }}$$$$f{}^{\prime }\left(x\right)=\frac{1}{\sqrt{\pi }}[\frac{\sin \left(\cos x\right)}{2\sqrt{x}}-\sqrt{x}\sin x\cos \left(\cos x\right)]$$$$f{}^{\prime }\left(\frac{\pi }{2}\right)=\frac{1}{\sqrt{\pi }}[0-\frac{\sqrt{\pi }}{\sqrt{2}}.1]=-\frac{\sqrt{2}}{2}$$

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