derivative of (1+csc(x^5)/(1-cot(x^5)))

解

\frac{d}{d x} (\frac{1 + csc ( x ^{5} )}{1 - cot ( x ^{5} )})

\frac{- 5 x ^{4} cot ( x ^{5} ) csc ( x ^{5} ) ( 1 - cot ( x ^{5} ) ) - 5 x ^{4} csc ^{2} ( x ^{5} ) ( 1 + csc ( x ^{5} ) )}{( 1 - cot ( x ^{5} ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 + csc ( x ^{5} )}{1 - cot ( x ^{5} )})

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 ^{5} ) ) ( 1 - cot ( x ^{5} ) ) - \frac{d}{d x} ( 1 - cot ( x ^{5} ) ) ( 1 + csc ( x ^{5} ) )}{( 1 - cot ( x ^{5} ) ) ^{2}}

\frac{d}{d x} (1 + csc (x^{5})) = - 5 x^{4} cot (x^{5}) csc (x^{5})

\frac{d}{d x} (1 - cot (x^{5})) = 5 x^{4} csc^{2} (x^{5})

= \frac{( - 5 x ^{4} cot ( x ^{5} ) csc ( x ^{5} ) ) ( 1 - cot ( x ^{5} ) ) - 5 x ^{4} csc ^{2} ( x ^{5} ) ( 1 + csc ( x ^{5} ) )}{( 1 - cot ( x ^{5} ) ) ^{2}}

簡素化

= \frac{- 5 x ^{4} cot ( x ^{5} ) csc ( x ^{5} ) ( 1 - cot ( x ^{5} ) ) - 5 x ^{4} csc ^{2} ( x ^{5} ) ( 1 + csc ( x ^{5} ) )}{( 1 - cot ( x ^{5} ) ) ^{2}}