(\partial)/(\partial x)(xyze^{x^2+y^2+z^2})

解

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

yz (e^{x^{2} + y^{2} + z^{2}} + 2 e^{x^{2} + y^{2} + z^{2}} x^{2})

解答ステップ

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

y, z

を定数として扱う

定数を除去:

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

= yz \frac{\partial}{\partial x} (x e^{x^{2} + y^{2} + z^{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} + z^{2}}

= yz (\frac{\partial}{\partial x} (x) e^{x^{2} + y^{2} + z^{2}} + \frac{\partial}{\partial x} (e^{x^{2} + y^{2} + z^{2}}) x)

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

\frac{\partial}{\partial x} (e^{x^{2} + y^{2} + z^{2}}) = e^{x^{2} + y^{2} + z^{2}} \cdot 2 x

= yz (1 \cdot e^{x^{2} + y^{2} + z^{2}} + e^{x^{2} + y^{2} + z^{2}} \cdot 2 xx)

簡素化

yz (1 \cdot e^{x^{2} + y^{2} + z^{2}} + e^{x^{2} + y^{2} + z^{2}} \cdot 2 xx) : yz (e^{x^{2} + y^{2} + z^{2}} + 2 e^{x^{2} + y^{2} + z^{2}} x^{2})

= yz (e^{x^{2} + y^{2} + z^{2}} + 2 e^{x^{2} + y^{2} + z^{2}} x^{2})