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

解答

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

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

求解步骤

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

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

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

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

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

化简

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

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

t an g e n t y = \frac{8}{x}, (16, \frac{8}{4})

\int_{0}^{1} 5 (1 + x)^{6} d x

x \to - 4 lim (3 x^{2} - 4 x + 8)

a re a cos (x), 2 - cos (x), [0, 2 π]

d er i v a t i v e - 16 t^{2} + 44 t + 31