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  • How to use it?

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

  • Identical expressions

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

  • Similar expressions

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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       log(x + 1)
f(x) = ----------
            2    
           x
f(x)=log⁡(x+1)x2f{\left(x \right)} = \frac{\log{\left(x + 1 \right)}}{x^{2}} 
f = log(x + 1)/x^2
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
log⁡(x+1)x2=0\frac{\log{\left(x + 1 \right)}}{x^{2}} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(x + 1)/x^2.
log⁡(1)02\frac{\log{\left(1 \right)}}{0^{2}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
1x2(x+1)−2log⁡(x+1)x3=0\frac{1}{x^{2} \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=46289.064371325x_{1} = 46289.064371325
x2=55915.3249294565x_{2} = 55915.3249294565
x3=43066.4423481692x_{3} = 43066.4423481692
x4=31173.3313715092x_{4} = 31173.3313715092
x5=41990.4768637283x_{5} = 41990.4768637283
x6=48433.3581910211x_{6} = 48433.3581910211
x7=50574.5841187331x_{7} = 50574.5841187331
x8=38756.8660046568x_{8} = 38756.8660046568
x9=30084.8495410589x_{9} = 30084.8495410589
x10=54848.509683824x_{10} = 54848.509683824
x11=53781.0457217524x_{11} = 53781.0457217524
x12=39835.7231631122x_{12} = 39835.7231631122
x13=36596.0007569018x_{13} = 36596.0007569018
x14=28994.9006573063x_{14} = 28994.9006573063
x15=37676.9721009352x_{15} = 37676.9721009352
x16=27903.4079776671x_{16} = 27903.4079776671
x17=26810.2872525746x_{17} = 26810.2872525746
x18=52712.9157330008x_{18} = 52712.9157330008
x19=32260.4163103634x_{19} = 32260.4163103634
x20=51644.1015564992x_{20} = 51644.1015564992
x21=45215.7074604315x_{21} = 45215.7074604315
x22=34430.6478336029x_{22} = 34430.6478336029
x23=33346.1687209482x_{23} = 33346.1687209482
x24=25715.4456218264x_{24} = 25715.4456218264
x25=35513.9083148437x_{25} = 35513.9083148437
x26=47361.606349444x_{26} = 47361.606349444
x27=44141.5093837038x_{27} = 44141.5093837038
x28=49504.3433661729x_{28} = 49504.3433661729
x29=40913.5815897579x_{29} = 40913.5815897579
The values of the extrema at the points:
(46289.06437132503, 5.01367225027522e-9)

(55915.32492945645, 3.49641108980673e-9)

(43066.442348169236, 5.75317647544992e-9)

(31173.33137150923, 1.06478695226975e-8)

(41990.47686372832, 6.03744388378113e-9)

(48433.35819102114, 4.5988617340066e-9)

(50574.58411873309, 4.2346049967001e-9)

(38756.86600465678, 7.03357070069416e-9)

(30084.849541058928, 1.13930295578254e-8)

(54848.50968382404, 3.62734250590277e-9)

(53781.04572175242, 3.76597001030357e-9)

(39835.72316311217, 6.67505555026526e-9)

(36596.0007569018, 7.84587348163993e-9)

(28994.90065730632, 1.22217885417782e-8)

(37676.97210093522, 7.42263360576165e-9)

(27903.407977667084, 1.31473640166017e-8)

(26810.28725257456, 1.41857245217561e-8)

(52712.91573300081, 3.91291776378939e-9)

(32260.416310363373, 9.97528922847023e-9)

(51644.10155649918, 4.06887511441187e-9)

(45215.7074604315, 5.24305727525894e-9)

(34430.64783360289, 8.81231556386405e-9)

(33346.16872094823, 9.36604094195435e-9)

(25715.44562182641, 1.53563125217244e-8)

(35513.9083148437, 8.30748079799758e-9)

(47361.60634944398, 4.79937759596204e-9)

(44141.50938370377, 5.48900567183577e-9)

(49504.34336617287, 4.41095362759701e-9)

(40913.581589757945, 6.34393187674682e-9)


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:
The function has no minima
The function has no maxima
Decreasing at the entire real axis
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
−1(x+1)2−4x(x+1)+6log⁡(x+1)x2x2=0\frac{- \frac{1}{\left(x + 1\right)^{2}} - \frac{4}{x \left(x + 1\right)} + \frac{6 \log{\left(x + 1 \right)}}{x^{2}}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=9102.30776252194x_{1} = 9102.30776252194
x2=10638.8166105699x_{2} = 10638.8166105699
x3=12424.7069464868x_{3} = 12424.7069464868
x4=8074.38528838515x_{4} = 8074.38528838515
x5=8331.66707889457x_{5} = 8331.66707889457
x6=7043.0027941782x_{6} = 7043.0027941782
x7=2861.66326372065x_{7} = 2861.66326372065
x8=9358.81248185136x_{8} = 9358.81248185136
x9=5487.78027628399x_{9} = 5487.78027628399
x10=12933.8259749235x_{10} = 12933.8259749235
x11=4705.46222167307x_{11} = 4705.46222167307
x12=4443.81949477837x_{12} = 4443.81949477837
x13=6266.77162202048x_{13} = 6266.77162202048
x14=13188.2132761766x_{14} = 13188.2132761766
x15=7301.20799885123x_{15} = 7301.20799885123
x16=11660.1189472527x_{16} = 11660.1189472527
x17=10383.1324543599x_{17} = 10383.1324543599
x18=8845.6202212781x_{18} = 8845.6202212781
x19=3391.80385636068x_{19} = 3391.80385636068
x20=11915.1074483123x_{20} = 11915.1074483123
x21=12679.3248882297x_{21} = 12679.3248882297
x22=5227.4110777538x_{22} = 5227.4110777538
x23=4181.6813396605x_{23} = 4181.6813396605
x24=7816.88817221548x_{24} = 7816.88817221548
x25=9871.3001468096x_{25} = 9871.3001468096
x26=6525.7990672492x_{26} = 6525.7990672492
x27=11404.999676375x_{27} = 11404.999676375
x28=2595.24233861184x_{28} = 2595.24233861184
x29=4966.64820972828x_{29} = 4966.64820972828
x30=8588.7425929889x_{30} = 8588.7425929889
x31=10127.2953833572x_{31} = 10127.2953833572
x32=3655.73120631204x_{32} = 3655.73120631204
x33=10894.3527982279x_{33} = 10894.3527982279
x34=5747.78179142154x_{34} = 5747.78179142154
x35=11149.7456838376x_{35} = 11149.7456838376
x36=7559.16594554901x_{36} = 7559.16594554901
x37=6784.53774379348x_{37} = 6784.53774379348
x38=3127.14533053197x_{38} = 3127.14533053197
x39=6007.43870139128x_{39} = 6007.43870139128
x40=3919.00282185223x_{40} = 3919.00282185223
x41=12169.9689280799x_{41} = 12169.9689280799
x42=9615.14116174033x_{42} = 9615.14116174033
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(−1(x+1)2−4x(x+1)+6log⁡(x+1)x2x2)=−∞\lim_{x \to 0^-}\left(\frac{- \frac{1}{\left(x + 1\right)^{2}} - \frac{4}{x \left(x + 1\right)} + \frac{6 \log{\left(x + 1 \right)}}{x^{2}}}{x^{2}}\right) = -\infty
lim⁡x→0+(−1(x+1)2−4x(x+1)+6log⁡(x+1)x2x2)=∞\lim_{x \to 0^+}\left(\frac{- \frac{1}{\left(x + 1\right)^{2}} - \frac{4}{x \left(x + 1\right)} + \frac{6 \log{\left(x + 1 \right)}}{x^{2}}}{x^{2}}\right) = \infty
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection point

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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(log⁡(x+1)x2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(log⁡(x+1)x2)=0\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{x^{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 log(x + 1)/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(x+1)xx2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(x+1)xx2)=0\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{x x^{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:
log⁡(x+1)x2=log⁡(1−x)x2\frac{\log{\left(x + 1 \right)}}{x^{2}} = \frac{\log{\left(1 - x \right)}}{x^{2}}
- No
log⁡(x+1)x2=−log⁡(1−x)x2\frac{\log{\left(x + 1 \right)}}{x^{2}} = - \frac{\log{\left(1 - x \right)}}{x^{2}}
- No
so, the function
not is
neither even, nor odd
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