derivative of (xsqrt(x+1)/(x+1))

解答

\frac{d}{d x} (\frac{x x + 1}{x + 1})

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

求解步骤

\frac{d}{d x} (\frac{x x + 1}{x + 1})

使用微分除法定则:

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

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

\frac{d}{d x} (x x + 1) = \frac{3 x + 2}{2 x + 1}

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

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

化简

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

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