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If $f(x)=\mid \cos x-\sin x\mid,then\;f'(\frac{\pi}{3})$=__________.

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Toolbox:
  • The derivative of a function $f(x)$ is $f'(x)$.
  • $f(x)=\cos x\Rightarrow f'(x)=-\sin x$
  • $f(x)=\sin x\Rightarrow f'(x)=\cos x$
  • $\sin\big(\large\frac{\pi}{3}\big)=\large\frac{\sqrt 3}{2}$
Step 1:
$f(x)=\mid \cos x-\sin x\mid$
$f'(x)=\mid-\sin x-\cos x\mid$
$\qquad=\mid -(\sin x+\cos x)\mid$
$f'(\large\frac{\pi}{3})=\mid $$\sin\large\frac{\pi}{3}+$$\cos \large\frac{\pi}{3}\mid$---(1)
Step 2:
$\sin\large\frac{\pi}{3}=\frac{\sqrt 3}{2}$ and $\cos\large\frac{\pi}{3}=\frac{1}{2}$
Substituting these in equ(1) we get,
$\Rightarrow \large\frac{\sqrt 3}{2}+\frac{1}{2}$
$\Rightarrow \bigg| \large\frac{\sqrt 3+1}{2}\bigg|$
answered Jul 4, 2013 by sreemathi.v
 
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