derivative of (3cos(x)/(1-sin(x)))

解

\frac{d}{d x} (\frac{3 cos ( x )}{1 - sin ( x )})

\frac{3}{1 - sin ( x )}

解答ステップ

\frac{d}{d x} (\frac{3 cos ( x )}{1 - sin ( x )})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (\frac{cos ( x )}{1 - sin ( x )})

商の法則を適用:

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

= 3 \cdot \frac{\frac{d}{d x} ( cos ( x ) ) ( 1 - sin ( x ) ) - \frac{d}{d x} ( 1 - sin ( x ) ) cos ( x )}{( 1 - sin ( x ) ) ^{2}}

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

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

= 3 \cdot \frac{( - sin ( x ) ) ( 1 - sin ( x ) ) - ( - cos ( x ) ) cos ( x )}{( 1 - sin ( x ) ) ^{2}}

簡素化

3 \cdot \frac{( - sin ( x ) ) ( 1 - sin ( x ) ) - ( - cos ( x ) ) cos ( x )}{( 1 - sin ( x ) ) ^{2}} : \frac{3}{1 - sin ( x )}

= \frac{3}{1 - sin ( x )}