A while ago, I learned how to take the derivative of $y=x^x$ using implicit differentiation, and I wondered if the same trick would work on every function of this type. I tried to differentiate $y=x^{x^x}$ the same way:

$\ln y=x^x\ln x$

$\ln (\ln y)=x \ln x+\ln (\ln x)$

$\frac{\frac{dy}{dx}}{y\ln y}=1+\ln x+\frac{1}{x\ln x}$

$\frac{dy}{dx}=x^{x^x+x}\ln x(1+\ln x+\frac{1}{x\ln x})$

This result seems to imply that all functions of this type, no matter how complicated, could be differentiated in this manner. My question is: Are there any functions of this sort that are so complicated that they are impossible to differentiate? If not, can this be proven?