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}}