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

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

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

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

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

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

簡素化

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

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