derivative of (x-cos(x)/(x^2+cot(x)))

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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