derivative of (ax-4sqrt(1+x)/(x^2-1))

解

\frac{d}{d x} (\frac{a x - 4 1 + x}{x ^{2} - 1})

\frac{- a x ^{2} + 6 x x + 1 + 2 x + 1 - a}{( x ^{2} - 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{a x - 4 1 + x}{x ^{2} - 1})

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 - 4 1 + x ) ( x ^{2} - 1 ) - \frac{d}{d x} ( x ^{2} - 1 ) ( a x - 4 1 + x )}{( x ^{2} - 1 ) ^{2}}

\frac{d}{d x} (a x - 4 1 + x) = a - \frac{2}{1 + x}

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

= \frac{( a - \frac{2}{1 + x} ) ( x ^{2} - 1 ) - 2 x ( a x - 4 1 + x )}{( x ^{2} - 1 ) ^{2}}

拡張

(a - \frac{2}{1 + x}) (x^{2} - 1) - 2 x (a x - 4 1 + x) : - a x^{2} + 6 x x + 1 + 2 x + 1 - a

= \frac{- a x ^{2} + 6 x x + 1 + 2 x + 1 - a}{( x ^{2} - 1 ) ^{2}}