d/dx(2/(x-3)-2ln(x-3)) - Wolfram|Alpha

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Derivative

d/dx(2/(x - 3) - 2 log(x - 3)) = -(2 (x - 2))/(x - 3)^2

Plot

Plot

Plot

Expanded form

4/(x^2 - 6 x + 9) - (2 x)/(x^2 - 6 x + 9)

(4 - 2 x)/(x^2 - 6 x + 9)

4/(x - 3)^2 - (2 x)/(x - 3)^2

Partial fraction expansion

-2/(x - 3) - 2/(x - 3)^2

Alternate form

(4 - 2 x)/((x - 6) x + 9)

Root

x = 2

Properties as a real function

Domain

{x element R : x!=3}

Range

{y element R : y<=1/2}

Series expansion at x=0

4/9 + (2 x)/27 - (2 x^3)/243 - (4 x^4)/729 + O(x^5)
(Taylor series)

Big‐O notation »

Series expansion at x=∞

-2/x - 8/x^2 - 30/x^3 - 108/x^4 + O((1/x)^5)
(Laurent series)

Big‐O notation »

Indefinite integral

integral-(2 (-2 + x))/(-3 + x)^2 dx = 2/(x - 3) - 2 log(x - 3) + constant
(assuming a complex-valued logarithm)

Global maximum

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

Limit

lim_(x-> ± ∞)-(2 (-2 + x))/(-3 + x)^2 = 0

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