Question

$If\left(x\right)=|\cos x-\sin x|,$ then find $f^{\prime }\left(\frac{\pi }{3}\right)$

Answer 1

Given $f\left(x\right)=|\cos x-\sin x|$

For $0<x<\frac{\pi }{4},f\left(x\right)=\cos x-\sin x$

For $\frac{\pi }{4}<x<\frac{\pi }{2},f\left(x\right)=\sin x-\cos x$

On differentiating

For $0<x<\frac{\pi }{4},f^{\prime }\left(x\right)=-\sin x-\cos x$

For $\frac{\pi }{4}<x<\frac{\pi }{2},f^{\prime }\left(x\right)=\cos x+\sin x$

$⟹f^{\prime }\left(\frac{\pi }{3}\right)=\cos \frac{\pi }{3}+\sin \frac{\pi }{3}=\frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{\sqrt{3}+1}{2}$