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

解

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

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

解答ステップ

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

x

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

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

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

\frac{\partial}{\partial y} (y^{2}) = 2 y

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

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