How do I solve [ln(x)]^(ln(x))?
I arrived at e^[ln(x)ln[lx(x)] . [1/x . ln[ln(x)]+ln(x).(1/(xln(x))]
You can't SOLVE anything as you don't have an EQUATION.
However, if as your title suggests, you are trying to find the DERIVATIVE of $\displaystyle \displaystyle \begin{align*} y = \left[ \ln{(x)} \right] ^{\ln{(x)}} \end{align*}$, then start by taking the logarithm of both sides and simplify the RHS before differentiating.
I spent 4 hrs on this question. The reason why I don't like typing equations using keyboard is because it's slow and frustrating, especially when the equation becomes long and chaotic.
Simply put, I utilized ln, e and the chain rule to differentiate the RHS. Answer turned out very differently, however. Using the same mathematical reasoning, I have no issues with the other question. It's probably a matter of algebraic simplification-hence, I want the confirmation.
If that is true-a matter of simplication-then, I'll skip past the question as I never bother with the rudimentary concrete details.
I applied chain rule to y = e^[ln(x)ln(ln(x)] and I'm very sure I have to. If you could tell me if this part is correct, it is sufficient as the steps beyond the chain rules are tautological simplication.