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

解

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

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

解答ステップ

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

簡素化

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

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{\frac{- x ^{2} - y ^{2}}{2}}, g = x y

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

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

\frac{d}{d x} (x y) = y + x \frac{d y}{d x}

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