(\partial)/(\partial x)(xy((8-xy)/(x+y)))

解

\frac{\partial}{\partial x} (x y (\frac{8 - x y}{x + y}))

\frac{y ( - x ^{2} y - 2 x y ^{2} + 8 y )}{( x + y ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (x y (\frac{8 - x y}{x + y}))

y

を定数として扱う

定数を除去:

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

= y \frac{\partial}{\partial x} (x \frac{8 - x y}{x + y})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = \frac{8 - x y}{x + y}

= y (\frac{\partial}{\partial x} (x) \frac{8 - x y}{x + y} + \frac{\partial}{\partial x} (\frac{8 - x y}{x + y}) x)

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

\frac{\partial}{\partial x} (\frac{8 - x y}{x + y}) = \frac{- y ^{2} - 8}{( x + y ) ^{2}}

= y (1 \cdot \frac{8 - x y}{x + y} + \frac{- y ^{2} - 8}{( x + y ) ^{2}} x)

簡素化

y (1 \cdot \frac{8 - x y}{x + y} + \frac{- y ^{2} - 8}{( x + y ) ^{2}} x) : \frac{y ( - x ^{2} y - 2 x y ^{2} + 8 y )}{( x + y ) ^{2}}

= \frac{y ( - x ^{2} y - 2 x y ^{2} + 8 y )}{( x + y ) ^{2}}