derivative of (ax+b/(ax-b))

解

\frac{d}{d x} (\frac{a x + b}{a x - b})

- \frac{2 ab}{( a x - b ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{a x + b}{a x - b})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( a x + b ) ( a x - b ) - \frac{d}{d x} ( a x - b ) ( a x + b )}{( a x - b ) ^{2}}

\frac{d}{d x} (a x + b) = a

\frac{d}{d x} (a x - b) = a

= \frac{a ( a x - b ) - a ( a x + b )}{( a x - b ) ^{2}}

簡素化

\frac{a ( a x - b ) - a ( a x + b )}{( a x - b ) ^{2}} : - \frac{2 ab}{( a x - b ) ^{2}}

= - \frac{2 ab}{( a x - b ) ^{2}}