derivative of ((1-x)/((x-1)^2+y^2))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (1 - x) = - 1

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

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

簡素化

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

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