derivative f(x)=(1+csc(x^3))/(1-cot(x^3))

解

導関数

f (x) = \frac{1 + csc ( x ^{3} )}{1 - cot ( x ^{3} )}

\frac{- 3 x ^{2} cot ( x ^{3} ) csc ( x ^{3} ) ( 1 - cot ( x ^{3} ) ) - 3 x ^{2} csc ^{2} ( x ^{3} ) ( 1 + csc ( x ^{3} ) )}{( 1 - cot ( x ^{3} ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 + csc ( x ^{3} )}{1 - cot ( x ^{3} )})

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 ^{3} ) ) ( 1 - cot ( x ^{3} ) ) - \frac{d}{d x} ( 1 - cot ( x ^{3} ) ) ( 1 + csc ( x ^{3} ) )}{( 1 - cot ( x ^{3} ) ) ^{2}}

\frac{d}{d x} (1 + csc (x^{3})) = - 3 x^{2} cot (x^{3}) csc (x^{3})

\frac{d}{d x} (1 - cot (x^{3})) = 3 x^{2} csc^{2} (x^{3})

= \frac{( - 3 x ^{2} cot ( x ^{3} ) csc ( x ^{3} ) ) ( 1 - cot ( x ^{3} ) ) - 3 x ^{2} csc ^{2} ( x ^{3} ) ( 1 + csc ( x ^{3} ) )}{( 1 - cot ( x ^{3} ) ) ^{2}}

簡素化

= \frac{- 3 x ^{2} cot ( x ^{3} ) csc ( x ^{3} ) ( 1 - cot ( x ^{3} ) ) - 3 x ^{2} csc ^{2} ( x ^{3} ) ( 1 + csc ( x ^{3} ) )}{( 1 - cot ( x ^{3} ) ) ^{2}}