derivative of (x+3)/((x+3)^2+4) - Wolfram|Alpha

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Derivative

d/dx((x + 3)/((x + 3)^2 + 4)) = -(x^2 + 6 x + 5)/(x^2 + 6 x + 13)^2

Plot

Plot

Plot

Alternate form

-((x + 1) (x + 5))/(x (x + 6) + 13)^2

((-x - 6) x - 5)/(x (x (x (x + 12) + 62) + 156) + 169)

Expanded form

-x^2/(x^4 + 12 x^3 + 62 x^2 + 156 x + 169) - (6 x)/(x^4 + 12 x^3 + 62 x^2 + 156 x + 169) - 5/(x^4 + 12 x^3 + 62 x^2 + 156 x + 169)

(-x^2 - 6 x - 5)/(x^4 + 12 x^3 + 62 x^2 + 156 x + 169)

Partial fraction expansion

8/(x^2 + 6 x + 13)^2 - 1/(x^2 + 6 x + 13)

Roots

x = -5

x = -1

Properties as a real function

Domain

R (all real numbers)

Range

{y element R : -1/32<=y<=1/4}

Series expansion at x=0

-5/169 - (18 x)/2197 + (357 x^2)/28561 - (2388 x^3)/371293 + (10175 x^4)/4826809 + O(x^5)
(Taylor series)

Big‐O notation »

Series expansion at x=∞

-(1/x)^2 + 6/x^3 - 15/x^4 - 36/x^5 + O((1/x)^6)
(Laurent series)

Big‐O notation »

Indefinite integral

integral-(5 + 6 x + x^2)/(13 + 6 x + x^2)^2 dx = (x + 3)/(x^2 + 6 x + 13) + constant

Global minima

$min{-(x^2 + 6 x + 5)/(x^2 + 6 x + 13)^2} = -1/32 at x = 2 sqrt(3) - 3$

$min{-(x^2 + 6 x + 5)/(x^2 + 6 x + 13)^2} = -1/32 at x = -3 - 2 sqrt(3)$

Global maximum

$max{-(x^2 + 6 x + 5)/(x^2 + 6 x + 13)^2} = 1/4 at x = -3$

Limit

lim_(x-> ± ∞)-(5 + 6 x + x^2)/(13 + 6 x + x^2)^2 = 0

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