derivative of e^{-(x^2+y^2})

解

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

e^{- x^{2} - y^{2}} (- 2 x - 2 y \frac{d y}{d x})

解答ステップ

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

簡素化

e^{- (x^{2} + y^{2})} : e^{- x^{2} - y^{2}}

= \frac{d}{d x} (e^{- x^{2} - y^{2}})

連鎖法則を適用:

e^{- x^{2} - y^{2}} \frac{d}{d x} (- x^{2} - y^{2})

= e^{- x^{2} - y^{2}} \frac{d}{d x} (- x^{2} - y^{2})

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

= e^{- x^{2} - y^{2}} (- 2 x - 2 y \frac{d y}{d x})