Question

How do you differentiate \[y = {\left( {\ln x} \right)^{\left( {\ln x} \right)}}\]?

Answer 1

Hint: In the given question, we have been given a logarithmic function. It has a base of Euler’s number, hence is the natural log. It has a natural log function raised to the power of the same, equal natural log function. We have to find the derivative of the given function. To do that, we need to know the function proper derivative of the given function.

Formula Used:
We are going to use the formula of first-order differentiation, which is,
\[\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}\]

Complete step by step answer:
We have to differentiate \[y = {\left( {\ln x} \right)^{\left( {\ln x} \right)}}\].
We are going to use the formula of first-order differentiation, which is,
\[\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}\]
\[\dfrac{{d\left( {{{\left( {\ln x} \right)}^{\left( {\ln x} \right)}}} \right)}}{{dx}} = \dfrac{{\left( {\ln x} \right).{{\left( {\ln x} \right)}^{\left( {\ln x} \right) - 1}}}}{x} = \dfrac{{{{\left( {\ln x} \right)}^{\left( {\ln x} \right)}}}}{x}\]

Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _b}a = n \Rightarrow {b^n} = a\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]
\[{\log _x}{x^n} = n\].

Note: In the given question, we had to find the derivative of a natural log function. For finding that, we only need to know the result we get after differentiating the natural log function, which is a standard result. So, it is really important that we know the formulae and where, when, and how to use them so that we can get the correct result.