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

解

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

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

解答ステップ

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

y

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

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

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

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

簡素化

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

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