(\partial)/(\partial x)(10xe^{-x^2-y^2})

解

\frac{\partial}{\partial x} (10 x e^{- x^{2} - y^{2}})

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

解答ステップ

\frac{\partial}{\partial x} (10 x e^{- x^{2} - y^{2}})

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{\partial}{\partial x} (x e^{- x^{2} - y^{2}})

積の法則を適用:

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

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

= 10 (\frac{\partial}{\partial x} (x) e^{- x^{2} - y^{2}} + \frac{\partial}{\partial x} (e^{- x^{2} - y^{2}}) x)

\frac{\partial}{\partial x} (x) = 1

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

= 10 (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} x) x)

簡素化

10 (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} x) x) : 10 (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})

= 10 (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})