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

解

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

2 e^{y^{2} x} y + 2 e^{y^{2} x} y^{3} x

解答ステップ

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

x

を定数として扱う

積の法則を適用:

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

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

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

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

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

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

e^{x y^{2}} \cdot 2 x y y^{2} = 2 e^{y^{2} x} y^{3} x

= 2 e^{y^{2} x} y + 2 e^{y^{2} x} y^{3} x