(\partial)/(\partial z)(e^{-x^2-2y^2-3z^2})

解

\frac{\partial}{\partial z} (e^{- x^{2} - 2 y^{2} - 3 z^{2}})

- 6 e^{- x^{2} - 2 y^{2} - 3 z^{2}} z

解答ステップ

\frac{\partial}{\partial z} (e^{- x^{2} - 2 y^{2} - 3 z^{2}})

x, y

を定数として扱う

連鎖法則を適用:

e^{- x^{2} - 2 y^{2} - 3 z^{2}} \frac{\partial}{\partial z} (- x^{2} - 2 y^{2} - 3 z^{2})

= e^{- x^{2} - 2 y^{2} - 3 z^{2}} \frac{\partial}{\partial z} (- x^{2} - 2 y^{2} - 3 z^{2})

\frac{\partial}{\partial z} (- x^{2} - 2 y^{2} - 3 z^{2}) = - 6 z

= e^{- x^{2} - 2 y^{2} - 3 z^{2}} (- 6 z)

簡素化

= - 6 e^{- x^{2} - 2 y^{2} - 3 z^{2}} z