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

解

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

e^{- x^{2} - y^{2}} y (1 - 2 x^{2})

解答ステップ

\frac{\partial}{\partial x} (\frac{x y}{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^{'}

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

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

= e^{- x^{2} - y^{2}} y (1 - 2 x^{2})