derivative 1/(x^2+1) - Wolfram|Alpha

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Derivative

d/dx(1/(x^2 + 1)) = -(2 x)/(x^2 + 1)^2

Plot

Plot

Plot

Expanded form

-(2 x)/(x^4 + 2 x^2 + 1)

Alternate form

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

Root

x = 0

Properties as a real function

Domain

R (all real numbers)

Range

{y element R : -(3 sqrt(3))/8<=y<=(3 sqrt(3))/8}

Parity

odd

Series expansion at x=0

-2 x + 4 x^3 - 6 x^5 + O(x^6)
(Taylor series)

Big‐O notation »

Series expansion at x=∞

-2/x^3 + 4/x^5 + O((1/x)^7)
(Laurent series)

Big‐O notation »

Indefinite integral

integral-(2 x)/(1 + x^2)^2 dx = 1/(x^2 + 1) + constant

Global minimum

$min{-(2 x)/(x^2 + 1)^2} = -(3 sqrt(3))/8 at x = 1/sqrt(3)$

Global maximum

$max{-(2 x)/(x^2 + 1)^2} = (3 sqrt(3))/8 at x = -1/sqrt(3)$

Limit

lim_(x-> ± ∞)-(2 x)/(1 + x^2)^2 = 0

Definite integral

integral_0^∞-(2 x)/(1 + x^2)^2 dx = -1

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