Tetration graph for this function

Question

I'm trying to visualize the derivatives of exponential tetration, by taking the original equation from its graph. Right now there is no elementary way of expressing the derivative. I'm not allowed to download computer sodftware, because my parents, worry this could slow the computer

Usually the equation should be odd:

May I have a graph of 2^^x, with 2 as the base, and x is the number of tetrations, with

x is -4 to 4 interval 1

y is -100 to 100 On the interval 1, because I don't want the image to be distorted on my "Desmos graphing calculator". This CANNOT however graph tetrations.

Answer 1

This is base2 Tetration, using Kneser's method, which is analytic in the upper and lower halves of the complex plane. The conjecture is that Kneser's solution is the only one with derivative>0 for z>-2, and which is analytic in the upper and lower halves of the complex plane, and at the real axis for z>-2. It has singularities at negative integers<=-2. Below is the Taylor series evaluated at z=0. The graph below goes from -1.9 to 3. Tet2(3.49090471067047)~=100.

{tet2=  1
+x^ 1*  0.889364954620976
+x^ 2*  0.00867654896536993
+x^ 3*  0.0952388000751818
+x^ 4* -0.00575234854012612
+x^ 5*  0.0129665820200372
+x^ 6* -0.00219604962303099
+x^ 7*  0.00199674684791144
+x^ 8* -0.000563354814878522
+x^ 9*  0.000348242328188164
+x^10* -0.000128532441264720
+x^11*  0.0000670819244205308
+x^12* -0.0000282987528227980
+x^13*  0.0000138001319906329
+x^14* -0.00000620190939837452
+x^15*  0.00000295556146480966
+x^16* -0.00000136867922453470
+x^17*  0.000000649057075651896
+x^18* -0.000000305166939328926
+x^19*  0.000000144948206151230
+x^20* -0.0000000687466431137918
+x^21*  0.0000000327674451778934
+x^22* -0.0000000156310874679970
+x^23*  0.00000000747812838918081
+x^24* -0.00000000358267681323948
+x^25*  0.00000000171986774579521
+x^26* -0.000000000826815962468010
+x^27*  0.000000000398110468382690
+x^28* -1.91942992587732 E-10
+x^29*  9.26631867862980 E-11
+x^30* -4.47869882913349 E-11
+x^31*  2.16712032119157 E-11
+x^32* -1.04969742914240 E-11
+x^33*  5.08944818189222 E-12
+x^34* -2.46987831946185 E-12
+x^35*  1.19965565647417 E-12
+x^36* -5.83165889022053 E-13
+x^37*  2.83702361814325 E-13
+x^38* -1.38118252499282 E-13
+x^39*  6.72883824735072 E-14
+x^40* -3.28030864119803 E-14
+x^41*  1.60015058245703 E-14
+x^42* -7.81025880698438 E-15
+x^43*  3.81431246327820 E-15
+x^44* -1.86381177420546 E-15
+x^45*  9.11196869462329 E-16
+x^46* -4.45694121263769 E-16
+x^47*  2.18105634006697 E-16
+x^48* -1.06780883356354 E-16
+x^49*  5.23008408468722 E-17
+x^50* -2.56274120196474 E-17
}

Answer 2

There's no agreed-upon definition of tetration for non-whole-number values (except that $^{-1}x=0$ for all $x$), so there's no continuity, hence no derivative. See the Wikipedia article for more on why this is.