(\partial)/(\partial x)(xye^{-8(x^2+y^2)})

解

\frac{\partial}{\partial x} (x y e^{- 8 (x^{2} + y^{2})})

y (e^{- 8 (x^{2} + y^{2})} - 16 e^{- 8 (x^{2} + y^{2})} x^{2})

解答ステップ

\frac{\partial}{\partial x} (x y e^{- 8 (x^{2} + y^{2})})

y

を定数として扱う

定数を除去:

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

= y \frac{\partial}{\partial x} (x e^{- 8 (x^{2} + y^{2})})

積の法則を適用:

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

f = x, g = e^{- 8 (x^{2} + y^{2})}

= y (\frac{\partial}{\partial x} (x) e^{- 8 (x^{2} + y^{2})} + \frac{\partial}{\partial x} (e^{- 8 (x^{2} + y^{2})}) x)

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

\frac{\partial}{\partial x} (e^{- 8 (x^{2} + y^{2})}) = - 16 e^{- 8 (x^{2} + y^{2})} x

= y (1 \cdot e^{- 8 (x^{2} + y^{2})} + (- 16 e^{- 8 (x^{2} + y^{2})} x) x)

簡素化

y (1 \cdot e^{- 8 (x^{2} + y^{2})} + (- 16 e^{- 8 (x^{2} + y^{2})} x) x) : y (e^{- 8 (x^{2} + y^{2})} - 16 e^{- 8 (x^{2} + y^{2})} x^{2})

= y (e^{- 8 (x^{2} + y^{2})} - 16 e^{- 8 (x^{2} + y^{2})} x^{2})