derivative of (1+sin(x)/(1+cos(x)))

解

\frac{d}{d x} (\frac{1 + sin ( x )}{1 + cos ( x )})

\frac{cos ( x ) ( 1 + cos ( x ) ) + sin ( x ) ( 1 + sin ( x ) )}{( 1 + cos ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 + sin ( x )}{1 + 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} ( 1 + sin ( x ) ) ( 1 + cos ( x ) ) - \frac{d}{d x} ( 1 + cos ( x ) ) ( 1 + sin ( x ) )}{( 1 + cos ( x ) ) ^{2}}

\frac{d}{d x} (1 + sin (x)) = cos (x)

\frac{d}{d x} (1 + cos (x)) = - sin (x)

= \frac{cos ( x ) ( 1 + cos ( x ) ) - ( - sin ( x ) ) ( 1 + sin ( x ) )}{( 1 + cos ( x ) ) ^{2}}

簡素化

= \frac{cos ( x ) ( 1 + cos ( x ) ) + sin ( x ) ( 1 + sin ( x ) )}{( 1 + cos ( x ) ) ^{2}}