derivative of 1/(3^{(-x^2-y^2)+1})

解

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

- \frac{ln ( 3 ) \cdot 3 ^{- x^{2} - y^{2}} ( - 2 x - 2 y \frac{d y}{d x} )}{( 3 ^{- x^{2} - y^{2}} + 1 ) ^{2}}

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

\frac{d}{d x} (3^{- x^{2} - y^{2}} + 1) = ln (3) \cdot 3^{- x^{2} - y^{2}} (- 2 x - 2 y \frac{d y}{d x})

= - \frac{1}{( 3 ^{- x^{2} - y^{2}} + 1 ) ^{2}} ln (3) \cdot 3^{- x^{2} - y^{2}} (- 2 x - 2 y \frac{d y}{d x})

簡素化

- \frac{1}{( 3 ^{- x^{2} - y^{2}} + 1 ) ^{2}} ln (3) \cdot 3^{- x^{2} - y^{2}} (- 2 x - 2 y \frac{d y}{d x}) : - \frac{ln ( 3 ) \cdot 3 ^{- x^{2} - y^{2}} ( - 2 x - 2 y \frac{d y}{d x} )}{( 3 ^{- x^{2} - y^{2}} + 1 ) ^{2}}

= - \frac{ln ( 3 ) \cdot 3 ^{- x^{2} - y^{2}} ( - 2 x - 2 y \frac{d y}{d x} )}{( 3 ^{- x^{2} - y^{2}} + 1 ) ^{2}}