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

解

導関数

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

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

解答ステップ

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

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

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

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

簡素化

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

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