با استفاده از فرمول ${{\left( {{u}^{n}} \right)}^{\prime }}=n{u}'.{{u}^{n-1}}$ ، خواهیم داشت:

$y\left( x \right)={{\cos }^{2}}\left( \frac{\pi }{3}+\frac{x}{4} \right)\Rightarrow {y}'\left( x \right)=2\times \underbrace{\left( -\frac{1}{4} \right)\sin \left( \frac{\pi }{3}+\frac{x}{4} \right)}_{{{u}'}}\underbrace{\cos \left( \frac{\pi }{3}+\frac{x}{4} \right)}_{u}$

با استفاده از اتحاد $\operatorname{sina}\times \operatorname{cosa}=\frac{1}{2}\sin 2a$ ، می‌توان نوشت:

$\Rightarrow {y}'\left( x \right)=-\frac{1}{2}\times \frac{1}{2}\sin \left( 2\left( \frac{\pi }{3}+\frac{x}{4} \right) \right)\Rightarrow {y}'\left( \frac{\pi }{3} \right)=-\frac{1}{2}\times \frac{1}{2}\sin \left( 2\times \frac{5\pi }{12} \right)\Rightarrow $ 

${y}'\left( \frac{\pi }{3} \right)=-\frac{1}{4}\sin \left( \frac{5\pi }{6} \right)=\frac{-1}{4}\sin \left( \pi -\frac{\pi }{6} \right)\Rightarrow {y}'\left( \frac{\pi }{3} \right)=\frac{-1}{4}\sin \frac{\pi }{6}=-\frac{1}{4}\times \frac{1}{2}=-\frac{1}{8}$