(\partial)/(\partial y)(-1/y sin(x/y))

解

\frac{\partial}{\partial y} (- \frac{1}{y} sin (\frac{x}{y}))

- \frac{- x cos ( \frac{x}{y} ) - y sin ( \frac{x}{y} )}{y ^{3}}

解答ステップ

\frac{\partial}{\partial y} (- \frac{1}{y} sin (\frac{x}{y}))

簡素化

- \frac{1}{y} sin (\frac{x}{y}) : - \frac{sin ( \frac{x}{y} )}{y}

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

x

を定数として扱う

定数を除去:

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

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

\frac{\partial}{\partial y} (y) = 1

= - \frac{( - \frac{x c o s ( \frac{x}{y} )}{y ^{2}} ) y - 1 \cdot sin ( \frac{x}{y} )}{y ^{2}}

簡素化

- \frac{( - \frac{x c o s ( \frac{x}{y} )}{y ^{2}} ) y - 1 \cdot sin ( \frac{x}{y} )}{y ^{2}} : - \frac{- x cos ( \frac{x}{y} ) - y sin ( \frac{x}{y} )}{y ^{3}}

= - \frac{- x cos ( \frac{x}{y} ) - y sin ( \frac{x}{y} )}{y ^{3}}