Question

Use logarithmic differentiation to find the derivative of the function.

55. \(y = {x^{lnx}}\)

Short answer

The answer is \(\frac{{dy}}{{dx}} = {x^{\ln x}}\left( {\frac{{2\ln x}}{x}} \right)\).

Step-by-step solution

Write the formula of the logarithmic differentiation and derivative of logarithmic functions

Derivatives of logarithmic functions:\(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)

Logarithmic differentiation:If\(y = {x^n}\)then\(\ln y = n\ln x\)

Finding the derivative of the function

Given that \(y = {x^{\ln x}}\).

Taking log both side to get,

\(\begin{array}{l}\ln y = \ln x\ln x\\\ln y = {\left( {\ln x} \right)^2}\end{array}\)

Now differentiating with respect to \(x\) we get,

\(\begin{array}{c}\frac{1}{y}\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\left( {\ln x} \right)^2}\\\frac{{dy}}{{dx}} = y\left( {2\frac{{\ln x}}{x}} \right)\\ = {x^{\ln x}}\frac{{2\ln x}}{x}\end{array}\)

Hence \(\frac{{dy}}{{dx}} = {x^{\ln x}}\left( {\frac{{2\ln x}}{x}} \right)\