derivative (x^3-x)/(1+x^2)

解答

导数

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

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

求解步骤

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

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

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

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

乘开

(3 x^{2} - 1) (1 + x^{2}) - 2 x (x^{3} - x) : x^{4} + 4 x^{2} - 1

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

t an g e n t f (x) = x^{2} - x, (- 4, 20)

x \cdot y \cdot \frac{d y}{d x} - y^{2} = 2 \cdot x^{2}

n = 4 \sum \infty \frac{1}{n ( n + 1 )}

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

\int_{0}^{2} \frac{1}{x + 3} d x