(a) If f ( x )=| tan 2 x |, then find the value of f^′( x ) at x =(π)/3.

$$\begin{aligned} & \mathrm{f}(\mathrm{x})=-\tan 2 \mathrm{x}, \frac{\pi}{4}<\mathrm{x}<\frac{\pi}{2} \\ & \mathrm{f}^{\prime}(\mathrm{x})=-2 \sec ^{2} 2 \mathrm{x}, \frac{\pi}{4}<\mathrm{x}<\frac{\pi}{2} \\ & \mathrm{f}^{\prime}\left(\frac{\pi}{3}\right)=-2(-2)^{2}=-8 \end{aligned} $$ $$\begin{aligned} & \mathrm{y}=\sqrt{1+\cot ^{2}\left(\cot ^{-1} \mathrm{x}\right)}=\sqrt{1+\mathrm{x}^{2}} \\ & \Rightarrow \frac{d \mathrm{y}}{d \mathrm{x}}=\frac{\mathrm{x}}{\sqrt{1+\mathrm{x}^{2}}} \\ & \Rightarrow \sqrt{1+\mathrm{x}^{2}} \frac{d \mathrm{y}}{d \mathrm{x}}-\mathrm{x}=0 \end{aligned} $$