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Question Number 136656 by 0731619177 last updated on 24/Mar/21

Answered by Ñï= last updated on 24/Mar/21

y=x^x^x^(...)   =x^y   y′=(e^(ylnx) )′=x^y (y′lnx+(y/x))=x^y y′lnx+yx^(y−1)   ⇒y′(1−x^y lnx)=yx^(y−1)   ⇒y′=((yx^(y−1) )/(1−x^y lnx))=((x^x^x^(...)   x^(x^x^x^(....)   −1) )/(1−x^x^x^x^(....)    lnx))

$${y}={x}^{{x}^{{x}^{...} } } ={x}^{{y}} \\ $$$${y}'=\left({e}^{{ylnx}} \right)'={x}^{{y}} \left({y}'{lnx}+\frac{{y}}{{x}}\right)={x}^{{y}} {y}'{lnx}+{yx}^{{y}−\mathrm{1}} \\ $$$$\Rightarrow{y}'\left(\mathrm{1}−{x}^{{y}} {lnx}\right)={yx}^{{y}−\mathrm{1}} \\ $$$$\Rightarrow{y}'=\frac{{yx}^{{y}−\mathrm{1}} }{\mathrm{1}−{x}^{{y}} {lnx}}=\frac{{x}^{{x}^{{x}^{...} } } {x}^{{x}^{{x}^{{x}^{....} } } −\mathrm{1}} }{\mathrm{1}−{x}^{{x}^{{x}^{{x}^{....} } } } {lnx}} \\ $$

Commented by 0731619177 last updated on 24/Mar/21

thanks

$${thanks} \\ $$

Commented by snipers237 last updated on 24/Mar/21

Sir why not  having    y=x^x^y   ? or  y=x^x^x^y    ?  that unfinished expression can′t  defined a real or finite number.

$${Sir}\:{why}\:{not}\:\:{having}\:\:\:\:{y}={x}^{{x}^{{y}} } \:?\:{or}\:\:{y}={x}^{{x}^{{x}^{{y}} } } \:? \\ $$$${that}\:{unfinished}\:{expression}\:{can}'{t}\:\:{defined}\:{a}\:{real}\:{or}\:{finite}\:{number}. \\ $$

Commented by MJS_new last updated on 25/Mar/21

y=x^x^(x...)    ln y =ln x^x^(x...)    ln y =x^x^(x...)  ln x  ln y =yln x  which can be solved for 0<x≤1

$${y}={x}^{{x}^{{x}...} } \\ $$$$\mathrm{ln}\:{y}\:=\mathrm{ln}\:{x}^{{x}^{{x}...} } \\ $$$$\mathrm{ln}\:{y}\:={x}^{{x}^{{x}...} } \mathrm{ln}\:{x} \\ $$$$\mathrm{ln}\:{y}\:={y}\mathrm{ln}\:{x} \\ $$$$\mathrm{which}\:\mathrm{can}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{for}\:\mathrm{0}<{x}\leqslant\mathrm{1} \\ $$

Commented by snipers237 last updated on 24/Mar/21

is  y  a real number? Does it exist as a finite number?  the logarithm properties hold only for positive real numbers  i know y can be defined as lim_(n→∞)  u_n   with u_0 =x>0  and  u_(n+1) =x^u_n       hen that converges.Aren′t there values of x>0  whose lead (u_n ) to diverge?

$${is}\:\:{y}\:\:{a}\:{real}\:{number}?\:{Does}\:{it}\:{exist}\:{as}\:{a}\:{finite}\:{number}? \\ $$$${the}\:{logarithm}\:{properties}\:{hold}\:{only}\:{for}\:{positive}\:{real}\:{numbers} \\ $$$${i}\:{know}\:{y}\:{can}\:{be}\:{defined}\:{as}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{u}_{{n}} \:\:{with}\:{u}_{\mathrm{0}} ={x}>\mathrm{0}\:\:{and}\:\:{u}_{{n}+\mathrm{1}} ={x}^{{u}_{{n}} } \:\:\: \\ $$$${hen}\:{that}\:{converges}.{Aren}'{t}\:{there}\:{values}\:{of}\:{x}>\mathrm{0}\:\:{whose}\:{lead}\:\left({u}_{{n}} \right)\:{to}\:{diverge}? \\ $$

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