derivative of x(dy/(dx))

解

\frac{d}{d x} (x \frac{d y}{d x})

\frac{d y}{d x} + x \frac{d ^{2} y}{d x ^{2}}

解答ステップ

\frac{d}{d x} (x \frac{d y}{d x})

積の法則を適用:

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

f = x, g = \frac{d y}{d x}

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

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

= 1 \frac{d y}{d x} + \frac{d ^{2} y}{d x ^{2}} x

簡素化

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