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

解

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

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

解答ステップ

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

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

\frac{d}{d x} (1 + cot (x)) = - csc^{2} (x)

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

簡素化

\frac{( - sin ( x ) ) ( 1 + cot ( x ) ) - ( - csc ^{2} ( x ) ) cos ( x )}{( 1 + cot ( x ) ) ^{2}} : \frac{- cos ( x ) - sin ( x ) + cot ( x ) csc ( x )}{( 1 + cot ( x ) ) ^{2}}

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