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

解

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

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

解答ステップ

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

x

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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