(\partial)/(\partial y)((-y)/(x^2+9y^2))

解

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

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

解答ステップ

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

簡素化

\frac{- y}{x ^{2} + 9 y ^{2}} : - \frac{y}{x ^{2} + 9 y ^{2}}

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

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{y}{x ^{2} + 9 y ^{2}})

商の法則を適用:

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

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

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

\frac{\partial}{\partial y} (x^{2} + 9 y^{2}) = 18 y

= - \frac{1 \cdot ( x ^{2} + 9 y ^{2} ) - 18 yy}{( x ^{2} + 9 y ^{2} ) ^{2}}

1 \cdot (x^{2} + 9 y^{2}) - 18 yy = x^{2} - 9 y^{2}

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