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

解答

导数

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

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

求解步骤

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

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \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}}

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

\frac{d x}{d x} = 1

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

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

化简

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

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

\frac{d ^{2}}{d x ^{2}} (6 x)

x \to 0 - lim (1)

n = 1 \sum \infty ln (n + 1)

d er i v a t i v e f (x) = 8 x^{2}

\frac{d}{d x} (ln (\frac{x ^{2} - 81}{k}))