(\partial)/(\partial x)((-x-6y)/(y^2+5x^2))

解

\frac{\partial}{\partial x} (\frac{- x - 6 y}{y ^{2} + 5 x ^{2}})

\frac{- y ^{2} + 5 x ^{2} + 60 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{- x - 6 y}{y ^{2} + 5 x ^{2}})

y

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 x} ( - x - 6 y ) ( y ^{2} + 5 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 5 x ^{2} ) ( - x - 6 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (- x - 6 y) = - 1

\frac{\partial}{\partial x} (y^{2} + 5 x^{2}) = 10 x

= \frac{( - 1 ) ( y ^{2} + 5 x ^{2} ) - 10 x ( - x - 6 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

簡素化

\frac{( - 1 ) ( y ^{2} + 5 x ^{2} ) - 10 x ( - x - 6 y )}{( y ^{2} + 5 x ^{2} ) ^{2}} : \frac{- y ^{2} + 5 x ^{2} + 60 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

= \frac{- y ^{2} + 5 x ^{2} + 60 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}