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

解

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

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

解答ステップ

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

簡素化

e^{- \frac{x ^{2}}{2} - \frac{y ^{2}}{2}} : e^{\frac{- x ^{2} - y ^{2}}{2}}

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

連鎖法則を適用:

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

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

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

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