(\partial)/(\partial y)(y^2sin(x/y))

解

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

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

解答ステップ

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

x

を定数として扱う

積の法則を適用:

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

f = y^{2}, g = sin (\frac{x}{y})

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

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

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

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

簡素化

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

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