(\partial)/(\partial x)(ysin(y/x))

解

\frac{\partial}{\partial x} (y sin (\frac{y}{x}))

- \frac{y ^{2} cos ( \frac{y}{x} )}{x ^{2}}

解答ステップ

\frac{\partial}{\partial x} (y sin (\frac{y}{x}))

y

を定数として扱う

定数を除去:

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

= y \frac{\partial}{\partial x} (sin (\frac{y}{x}))

連鎖法則を適用:

cos (\frac{y}{x}) \frac{\partial}{\partial x} (\frac{y}{x})

= cos (\frac{y}{x}) \frac{\partial}{\partial x} (\frac{y}{x})

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

= y cos (\frac{y}{x}) (- \frac{y}{x ^{2}})

簡素化

y cos (\frac{y}{x}) (- \frac{y}{x ^{2}}) : - \frac{y ^{2} cos ( \frac{y}{x} )}{x ^{2}}

= - \frac{y ^{2} cos ( \frac{y}{x} )}{x ^{2}}