(\partial)/(\partial x)(((2+t^2))/(3x-x^3))

解

\frac{\partial}{\partial x} (\frac{( 2 + t ^{2} )}{3 x - x ^{3}})

- \frac{( 2 + t ^{2} ) ( 3 - 3 x ^{2} )}{( 3 x - x ^{3} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{( 2 + t ^{2} )}{3 x - x ^{3}})

t

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= (2 + t^{2}) \frac{\partial}{\partial x} (\frac{1}{3 x - x ^{3}})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= (2 + t^{2}) \frac{\partial}{\partial x} ((3 x - x^{3})^{- 1})

連鎖法則を適用:

- \frac{1}{( 3 x - x ^{3} ) ^{2}} \frac{\partial}{\partial x} (3 x - x^{3})

= - \frac{1}{( 3 x - x ^{3} ) ^{2}} \frac{\partial}{\partial x} (3 x - x^{3})

\frac{\partial}{\partial x} (3 x - x^{3}) = 3 - 3 x^{2}

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

簡素化

(2 + t^{2}) (- \frac{1}{( 3 x - x ^{3} ) ^{2}} (3 - 3 x^{2})) : - \frac{( 2 + t ^{2} ) ( 3 - 3 x ^{2} )}{( 3 x - x ^{3} ) ^{2}}

= - \frac{( 2 + t ^{2} ) ( 3 - 3 x ^{2} )}{( 3 x - x ^{3} ) ^{2}}