derivative of (1+csc(x^9)/(1-cot(x^9)))

解

\frac{d}{d x} (\frac{1 + csc ( x ^{9} )}{1 - cot ( x ^{9} )})

\frac{- 9 x ^{8} cot ( x ^{9} ) csc ( x ^{9} ) ( 1 - cot ( x ^{9} ) ) - 9 x ^{8} csc ^{2} ( x ^{9} ) ( 1 + csc ( x ^{9} ) )}{( 1 - cot ( x ^{9} ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 + csc ( x ^{9} )}{1 - cot ( x ^{9} )})

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 + csc ( x ^{9} ) ) ( 1 - cot ( x ^{9} ) ) - \frac{d}{d x} ( 1 - cot ( x ^{9} ) ) ( 1 + csc ( x ^{9} ) )}{( 1 - cot ( x ^{9} ) ) ^{2}}

\frac{d}{d x} (1 + csc (x^{9})) = - 9 x^{8} cot (x^{9}) csc (x^{9})

\frac{d}{d x} (1 - cot (x^{9})) = 9 x^{8} csc^{2} (x^{9})

= \frac{( - 9 x ^{8} cot ( x ^{9} ) csc ( x ^{9} ) ) ( 1 - cot ( x ^{9} ) ) - 9 x ^{8} csc ^{2} ( x ^{9} ) ( 1 + csc ( x ^{9} ) )}{( 1 - cot ( x ^{9} ) ) ^{2}}

簡素化

= \frac{- 9 x ^{8} cot ( x ^{9} ) csc ( x ^{9} ) ( 1 - cot ( x ^{9} ) ) - 9 x ^{8} csc ^{2} ( x ^{9} ) ( 1 + csc ( x ^{9} ) )}{( 1 - cot ( x ^{9} ) ) ^{2}}