Question

Find the derivatives of the following functions at the indicated points.
$f\left(x\right)=sin4xcos4x,f^{\prime }\left(\pi /3\right)=?$

Answer 1

$f\left(x\right)=\sin 4x\cos 4x=\frac{2\sin 4x\cos 4x}{2}$

$⟹f\left(x\right)=\frac{1}{2}\sin 8x$

$⟹f^{\prime }\left(x\right)=\frac{1}{2}.8\cos 8x=4\cos 8x$

$⟹f^{\prime }\left(\pi /3\right)=4\cos \left(8\pi /3\right)=4\cos (2\pi +2\pi /3)$

$⟹f^{\prime }\left(\pi /3\right)=4\cos \left(2\pi /3\right)=4.\frac{-1}{2}$

$⟹f^{\prime }\left(\pi /3\right)=-2$