Differentiate y=a^x^a^x^a----to infinity?
2 Answers
where
Explanation:
#y=a^(x^(a^(x^cdots)))#
Take the natural logarithm:
#lny=ln(a^(x^(a^(x^cdots))))#
#color(white)lny=x^(a^(x^(a^cdots)))lna#
Take the natural logarithm once more:
#ln(lny)=ln(x^(a^(x^(a^cdots)))lna)#
#color(white)ln(lny)=ln(x^(a^(x^(a^cdots))))+ln(lna)#
#color(white)ln(lny)=a^(x^(a^(x^cdots)))lnx+ln(lna)#
#color(white)ln(lny)=ylnx+ln(lna)#
Taking the derivative of this last version:
#d/dxln(lny)=d/dx(ylnx)+d/dxln(lna)#
Using the chain rule (left) and product rule (right):
#1/lny(d/dxlny)=dy/dxlnx+y/x+0#
#1/lny(1/y)dy/dx=dy/dxlnx+y/x#
Grouping
#dy/dx(1/(ylny)-lnx)=y/x#
#dy/dx((1-ylnxlny)/(ylny))=y/x#
#dy/dx=(y^2lny)/(x(1-ylnxlny))#
Explanation:
If
then