(\partial)/(\partial y)((xy+3x^2)/(1+y))

解

\frac{\partial}{\partial y} (\frac{x y + 3 x ^{2}}{1 + y})

\frac{x - 3 x ^{2}}{( 1 + y ) ^{2}}

解答ステップ

\frac{\partial}{\partial y} (\frac{x y + 3 x ^{2}}{1 + y})

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} ( x y + 3 x ^{2} ) ( 1 + y ) - \frac{\partial}{\partial y} ( 1 + y ) ( x y + 3 x ^{2} )}{( 1 + y ) ^{2}}

\frac{\partial}{\partial y} (x y + 3 x^{2}) = x

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

= \frac{x ( 1 + y ) - 1 \cdot ( x y + 3 x ^{2} )}{( 1 + y ) ^{2}}

簡素化

\frac{x ( 1 + y ) - 1 \cdot ( x y + 3 x ^{2} )}{( 1 + y ) ^{2}} : \frac{x - 3 x ^{2}}{( 1 + y ) ^{2}}

= \frac{x - 3 x ^{2}}{( 1 + y ) ^{2}}