(\partial)/(\partial y)((e^y)/(x+y^9))

解

\frac{\partial}{\partial y} (\frac{e ^{y}}{x + y ^{9}})

\frac{e ^{y} ( x + y ^{9} ) - 9 y ^{8} e ^{y}}{( x + y ^{9} ) ^{2}}

解答ステップ

\frac{\partial}{\partial y} (\frac{e ^{y}}{x + y ^{9}})

x

を定数として扱う

商の法則を適用:

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

= \frac{\frac{\partial}{\partial y} ( e ^{y} ) ( x + y ^{9} ) - \frac{\partial}{\partial y} ( x + y ^{9} ) e ^{y}}{( x + y ^{9} ) ^{2}}

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

\frac{\partial}{\partial y} (x + y^{9}) = 9 y^{8}

= \frac{e ^{y} ( x + y ^{9} ) - 9 y ^{8} e ^{y}}{( x + y ^{9} ) ^{2}}