**Toolbox:**

- before differentiating eliminate modulus by its definition.

$ | cos\: x|=cosx\: when \: x \in \bigg[ 0, \large\frac{\pi}{2} \bigg]$

$ | sin\: x|=sinx\: when \: x \in [ 0, \pi ]$

$ | cos\: x|=-cosx\: when \: x \in \bigg[\large \frac{\pi}{2},\large \frac{3\pi}{2} \bigg]$

$ | sin\: x|=-sinx\: when \: x \in [ \pi, 2\pi ]$

$ \large\frac{2\pi}{3} \in \bigg[\large \frac{\pi}{2}, \pi \bigg]$

$ \Rightarrow y= -cosx+sinx$

$ \large\frac{dy}{dx}=sinx+cosx$

$ = \large\frac{\sqrt 3}{2}-\large\frac{1}{2}=\large\frac{\sqrt 3-1}{2}\: at \: x=\large\frac{2\pi}{3}$