derivative of (sin(4x)/(2+cos(4x)))

解

\frac{d}{d x} (\frac{sin ( 4 x )}{2 + cos ( 4 x )})

\frac{4 cos ( 4 x ) ( 2 + cos ( 4 x ) ) + 4 sin ^{2} ( 4 x )}{( 2 + cos ( 4 x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{sin ( 4 x )}{2 + cos ( 4 x )})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( sin ( 4 x ) ) ( 2 + cos ( 4 x ) ) - \frac{d}{d x} ( 2 + cos ( 4 x ) ) sin ( 4 x )}{( 2 + cos ( 4 x ) ) ^{2}}

\frac{d}{d x} (sin (4 x)) = cos (4 x) \cdot 4

\frac{d}{d x} (2 + cos (4 x)) = - 4 sin (4 x)

= \frac{cos ( 4 x ) \cdot 4 ( 2 + cos ( 4 x ) ) - ( - 4 sin ( 4 x ) ) sin ( 4 x )}{( 2 + cos ( 4 x ) ) ^{2}}

簡素化

\frac{cos ( 4 x ) \cdot 4 ( 2 + cos ( 4 x ) ) - ( - 4 sin ( 4 x ) ) sin ( 4 x )}{( 2 + cos ( 4 x ) ) ^{2}} : \frac{4 cos ( 4 x ) ( 2 + cos ( 4 x ) ) + 4 sin ^{2} ( 4 x )}{( 2 + cos ( 4 x ) ) ^{2}}

= \frac{4 cos ( 4 x ) ( 2 + cos ( 4 x ) ) + 4 sin ^{2} ( 4 x )}{( 2 + cos ( 4 x ) ) ^{2}}