(\partial)/(\partial y)(e^{-(x^2+y^2)})

解

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

- 2 e^{- x^{2} - y^{2}} y

解答ステップ

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

簡素化

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

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

x

を定数として扱う

連鎖法則を適用:

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

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

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

= e^{- x^{2} - y^{2}} (- 2 y)

簡素化

= - 2 e^{- x^{2} - y^{2}} y