Question

help me pls

find Dx [x^x^x^x^x]

Answer 1

find \[Dx [x ^{x ^{x ^{x ^{x}}}}]\]

Answer 2

Oh my. Is that really how the question looks like? If so, sorry but my brain rejects it at the sight.

Answer 3

@wolfe8 it's ok :)

Answer 4

lol i agree wolfe

Answer 5

Idk does it like involve log or ln or something

Answer 6

@fgiancarelli help

Answer 7

For the first iteration, you have,
\[y=x^{x^{x^{x^x}}} \Rightarrow\ln y=x^{x^{x^{x}}}\ln x\] So, doing derivatives in both members,
\[y'/y=(x^{x^{x^{x}}})'\ln x+x^{x^{x^{x}}}/x\]
So, you see that you now will need the derivative of another function. To simplify, lets put the following nomenclature,
\[y_1=x^{x^{x^{x^x}}}\]\[y_2=x^{x^{x^{x}}}\]\[y_3=x^{x^{x}}\]\[y_4=x^{x}\]
And we proceed with the last one,
\[y'/y_4=\ln x+1\Rightarrow y'=y_4(\ln x+1)\]
Following the reasoning (from y_4 to y_1) you'll have,
\[y'=y_1(y_2(y_3y_4(\ln x+1)\ln x+x^x/x)\ln x+x^{x^{x}}/x)\ln x+x^{x^{x^{x}}}/x\]

Answer 8

Personally I wouldn't do the nomenclature :P .