(\partial)/(\partial x)(x^2y^2e^{4xy})

解

\frac{\partial}{\partial x} (x^{2} y^{2} e^{4 x y})

y^{2} (2 x e^{4 x y} + e^{4 x y} \cdot 4 y x^{2})

解答ステップ

\frac{\partial}{\partial x} (x^{2} y^{2} e^{4 x y})

y

を定数として扱う

定数を除去:

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

= y^{2} \frac{\partial}{\partial x} (x^{2} e^{4 x y})

積の法則を適用:

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

f = x^{2}, g = e^{4 x y}

= y^{2} (\frac{\partial}{\partial x} (x^{2}) e^{4 x y} + \frac{\partial}{\partial x} (e^{4 x y}) x^{2})

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

\frac{\partial}{\partial x} (e^{4 x y}) = e^{4 x y} \cdot 4 y

= y^{2} (2 x e^{4 x y} + e^{4 x y} \cdot 4 y x^{2})