# Logarithmic differentiation

• Sep 10th 2012, 09:09 AM
Nora314
Logarithmic differentiation
Use logarithmic differentiation to fi nd the derivative of y with respect the independent
variable x, where

y = xln x

thanks to everyone who takes a look :D
• Sep 10th 2012, 10:08 AM
Prove It
Re: Logarithmic differentiation
Quote:

Originally Posted by Nora314
Use logarithmic differentiation to find the derivative of y with respect the independent
variable x, where

y = xln x

thanks to everyone who takes a look :D

\displaystyle \begin{align*} y &= x^{\ln{x}} \\ \ln{y} &= \ln{\left( x^{\ln{x}} \right)} \\ \ln{y} &= \ln{x}\ln{x} \\ \ln{y} &= \left( \ln{x} \right)^2 \\ \frac{d}{dx} \left( \ln{y} \right) &= \frac{d}{dx} \left[ \left( \ln{x} \right)^2 \right] \\ \frac{d}{dy} \left( \ln{y} \right)\,\frac{dy}{dx} &= \frac{2\ln{x}}{x} \\ \frac{1}{y}\,\frac{dy}{dx} &= \frac{2\ln{x}}{x} \\ \frac{dy}{dx} &= \frac{2y\ln{x}}{x} \\ \frac{dy}{dx} &= \frac{2x^{\ln{x}} \ln{x} }{ x } \end{align*}
• Sep 10th 2012, 12:34 PM
Nora314
Re: Logarithmic differentiation
Thank you! :) makes sense now!