derivative of xe^{9xy}

解

\frac{d}{d x} (x e^{9 x y})

e^{9 x y} + 9 e^{9 x y} x (y + x \frac{d y}{d x})

解答ステップ

\frac{d}{d x} (x e^{9 x y})

積の法則を適用:

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

f = x, g = e^{9 x y}

= \frac{d x}{d x} e^{9 x y} + \frac{d}{d x} (e^{9 x y}) x

\frac{d x}{d x} = 1

\frac{d}{d x} (e^{9 x y}) = e^{9 x y} \cdot 9 (y + x \frac{d y}{d x})

= 1 \cdot e^{9 x y} + e^{9 x y} \cdot 9 (y + x \frac{d y}{d x}) x

簡素化

= e^{9 x y} + 9 e^{9 x y} x (y + x \frac{d y}{d x})