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

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

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

\frac{\partial}{\partial x} ((x^{2} - y^{2})^{2}) = 4 x (x^{2} - y^{2})

= 2 y \frac{1 \cdot ( x ^{2} - y ^{2} ) ^{2} - 4 x ( x ^{2} - y ^{2} ) x}{( ( x ^{2} - y ^{2} ) ^{2} ) ^{2}}

簡素化

2 y \frac{1 \cdot ( x ^{2} - y ^{2} ) ^{2} - 4 x ( x ^{2} - y ^{2} ) x}{( ( x ^{2} - y ^{2} ) ^{2} ) ^{2}} : \frac{2 y ( - 3 x ^{2} - y ^{2} )}{( x ^{2} - y ^{2} ) ^{3}}

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