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

解

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

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

解答ステップ

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

x

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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