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

解

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

\frac{- 2 a - 2 a x ^{2} - 3 x x + 1 - x + 1}{2 ( x ^{2} - 1 ) ^{2}}

解答ステップ

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

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

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

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

簡素化

\frac{( a + \frac{1}{2 1 + x} ) ( x ^{2} - 1 ) - 2 x ( a x + 1 + x )}{( x ^{2} - 1 ) ^{2}} : \frac{- 2 a - 2 a x ^{2} - 3 x x + 1 - x + 1}{2 ( x ^{2} - 1 ) ^{2}}

= \frac{- 2 a - 2 a x ^{2} - 3 x x + 1 - x + 1}{2 ( x ^{2} - 1 ) ^{2}}