2 cosx at x = 2 - Vidyadip

Question

2 cosx at x = $\frac{\pi}{2}$

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Answer

$$We have,
$\because\text{f(x)}=2\cos\text{x}$
$\therefore\text{f}'\text{(a)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{(a+h)}-\text{f(a)}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}\Big(\frac{\pi}{2}+\text{h}\Big)-\text{f}\Big(\frac{\pi}{2}\Big)}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}\Big(\frac{\pi}{2}+\text{h}\Big)-2\cos\frac{\pi}{2}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{-2\text{sinh}-0}{\text{h}}$
$=-2\ \Big[\because\lim_\limits{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$\therefore\text{f}'\Big(\frac{\pi}{2}\Big)=-2$

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