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

解

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

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

解答ステップ

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

x

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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