derivative of e^{xy}(y+x(dy/(dx)))

解

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

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

解答ステップ

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

積の法則を適用:

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

f = e^{x y}, g = y + x \frac{d y}{d x}

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

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

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

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

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

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