derivative of (ax+b/(cx+d))

解

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

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

解答ステップ

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

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

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

\frac{d}{d x} (c x + d) = c

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

拡張

a (c x + d) - c (a x + b) : - b c + a d

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