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

  • How to use it?

  • Graphing y =:
  • x^3/2(x+1)^2
  • x^3+6x^2+9x
  • 4x/(x^2+1)^2 4x/(x^2+1)^2
  • -x^4+2x^2+3
  • Derivative of:
  • 4x/(x^2+1)^2

  • Identical expressions

  • 4x/(x^ two + one)^ two
  • 4x divide by (x squared plus 1) squared
  • 4x divide by (x to the power of two plus one) to the power of two
  • 4x/(x2+1)2
  • 4x/x2+12
  • 4x/(x²+1)²
  • 4x/(x to the power of 2+1) to the power of 2
  • 4x/x^2+1^2
  • 4x divide by (x^2+1)^2

  • Similar expressions

  • 4x/(x^2-1)^2
Plotting a function graph/ 4x/(x^2+1)^2

Graphing y = 4x/(x^2+1)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          4*x   
f(x) = ---------
               2
       / 2    \ 
       \x  + 1/
f(x)=4x(x2+1)2f{\left(x \right)} = \frac{4 x}{\left(x^{2} + 1\right)^{2}} 
f = 4*x/((x^2 + 1)^2)
The graph of the function
050001000015000200002500030000350002.5-2.5
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
4x(x2+1)2=0\frac{4 x}{\left(x^{2} + 1\right)^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=28888.9499392717x_{1} = 28888.9499392717
x2=23804.3479553544x_{2} = 23804.3479553544
x3=39906.4824805783x_{3} = 39906.4824805783
x4=−29605.2051269265x_{4} = -29605.2051269265
x5=25499.1699170213x_{5} = 25499.1699170213
x6=24651.7522903592x_{6} = 24651.7522903592
x7=30583.8929111256x_{7} = 30583.8929111256
x8=42449.0875005062x_{8} = 42449.0875005062
x9=−27062.8315909668x_{9} = -27062.8315909668
x10=−28757.7386512816x_{10} = -28757.7386512816
x11=36516.3874515588x_{11} = 36516.3874515588
x12=15331.5208433594x_{12} = 15331.5208433594
x13=28041.4904927672x_{13} = 28041.4904927672
x14=22109.5852761254x_{14} = 22109.5852761254
x15=−24520.549154346x_{15} = -24520.549154346
x16=−42317.8645113938x_{16} = -42317.8645113938
x17=−38927.7323209959x_{17} = -38927.7323209959
x18=16178.6518916848x_{18} = 16178.6518916848
x19=−32147.6481435989x_{19} = -32147.6481435989
x20=31431.3751377026x_{20} = 31431.3751377026
x21=−39775.2608173139x_{21} = -39775.2608173139
x22=−22825.7598445709x_{22} = -22825.7598445709
x23=−21131.0378020927x_{23} = -21131.0378020927
x24=20414.8966506568x_{24} = 20414.8966506568
x25=−25367.9648198459x_{25} = -25367.9648198459
x26=−21978.3894419472x_{26} = -21978.3894419472
x27=18720.3022019605x_{27} = 18720.3022019605
x28=33126.3583237319x_{28} = 33126.3583237319
x29=38211.4278248492x_{29} = 38211.4278248492
x30=−16047.4885294021x_{30} = -16047.4885294021
x31=34821.3634468047x_{31} = 34821.3634468047
x32=21262.2305994173x_{32} = 21262.2305994173
x33=−23673.1469943384x_{33} = -23673.1469943384
x34=40754.0146073646x_{34} = 40754.0146073646
x35=−36385.1680059945x_{35} = -36385.1680059945
x36=32278.8637725529x_{36} = 32278.8637725529
x37=−35537.6544673896x_{37} = -35537.6544673896
x38=0x_{38} = 0
x39=−38080.207196563x_{39} = -38080.207196563
x40=−30452.6792719769x_{40} = -30452.6792719769
x41=−15200.3654382736x_{41} = -15200.3654382736
x42=29736.4176408609x_{42} = 29736.4176408609
x43=−26215.3926807132x_{43} = -26215.3926807132
x44=17025.8300696591x_{44} = 17025.8300696591
x45=19567.5861240751x_{45} = 19567.5861240751
x46=−17741.8726673034x_{46} = -17741.8726673034
x47=−31300.1604633294x_{47} = -31300.1604633294
x48=−40622.792474443x_{48} = -40622.792474443
x49=−34690.1453612251x_{49} = -34690.1453612251
x50=39058.9534836425x_{50} = 39058.9534836425
x51=−18589.1211308922x_{51} = -18589.1211308922
x52=−33842.6410205595x_{52} = -33842.6410205595
x53=−32995.141812677x_{53} = -32995.141812677
x54=−16894.6599134202x_{54} = -16894.6599134202
x55=17873.0486704095x_{55} = 17873.0486704095
x56=41601.5496726855x_{56} = 41601.5496726855
x57=−20283.7072775985x_{57} = -20283.7072775985
x58=−27910.2805438436x_{58} = -27910.2805438436
x59=35668.8732572414x_{59} = 35668.8732572414
x60=33973.8583483916x_{60} = 33973.8583483916
x61=22956.9583845x_{61} = 22956.9583845
x62=−41470.3270985658x_{62} = -41470.3270985658
x63=37363.9057313917x_{63} = 37363.9057313917
x64=−37232.6856743107x_{64} = -37232.6856743107
x65=26346.5995523547x_{65} = 26346.5995523547
x66=−19436.4006315048x_{66} = -19436.4006315048
x67=27194.0400733442x_{67} = 27194.0400733442
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 4*x/((x^2 + 1)^2).
4⋅0⋅1(02+1)24 \cdot 0 \cdot \frac{1}{\left(0^{2} + 1\right)^{2}}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−16x2(x2+1)3+4(x2+1)2=0- \frac{16 x^{2}}{\left(x^{2} + 1\right)^{3}} + \frac{4}{\left(x^{2} + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−33x_{1} = - \frac{\sqrt{3}}{3}
x2=33x_{2} = \frac{\sqrt{3}}{3}
The values of the extrema at the points:
    ___        ___  
 -\/ 3    -3*\/ 3   
(-------, ---------)
    3         4     

   ___      ___ 
 \/ 3   3*\/ 3  
(-----, -------)
   3       4


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−33x_{1} = - \frac{\sqrt{3}}{3}
Maxima of the function at points:
x1=33x_{1} = \frac{\sqrt{3}}{3}
Decreasing at intervals
[−33,33]\left[- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right]
Increasing at intervals
(−∞,−33]∪[33,∞)\left(-\infty, - \frac{\sqrt{3}}{3}\right] \cup \left[\frac{\sqrt{3}}{3}, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
16x(6x2x2+1−3)(x2+1)3=0\frac{16 x \left(\frac{6 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{3}} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
x2=0x_{2} = 0
x3=1x_{3} = 1

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[−1,0]∪[1,∞)\left[-1, 0\right] \cup \left[1, \infty\right)
Convex at the intervals
(−∞,−1]∪[0,1]\left(-\infty, -1\right] \cup \left[0, 1\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(4x(x2+1)2)=0\lim_{x \to -\infty}\left(\frac{4 x}{\left(x^{2} + 1\right)^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(4x(x2+1)2)=0\lim_{x \to \infty}\left(\frac{4 x}{\left(x^{2} + 1\right)^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 4*x/((x^2 + 1)^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(4(x2+1)2)=0\lim_{x \to -\infty}\left(\frac{4}{\left(x^{2} + 1\right)^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(4(x2+1)2)=0\lim_{x \to \infty}\left(\frac{4}{\left(x^{2} + 1\right)^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
4x(x2+1)2=−4x(x2+1)2\frac{4 x}{\left(x^{2} + 1\right)^{2}} = - \frac{4 x}{\left(x^{2} + 1\right)^{2}}
- No
4x(x2+1)2=4x(x2+1)2\frac{4 x}{\left(x^{2} + 1\right)^{2}} = \frac{4 x}{\left(x^{2} + 1\right)^{2}}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = 4x/(x^2+1)^2
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