The discussion focuses on using logarithmic differentiation to find the derivative of the function y = (1+x)^(1/x). The solution involves taking the natural logarithm of both sides, resulting in ln y = (1/x) ln(1+x). The derivative is calculated as dy/dx = y [(-1/x^2)(ln(1+x) + x/(1+x))], confirming the application of logarithmic differentiation techniques. The final expression for dy/dx incorporates both the original function and its logarithmic transformation.
PREREQUISITESStudents studying calculus, particularly those focusing on differentiation techniques, as well as educators looking for examples of logarithmic differentiation applications.
chapsticks said:Homework Statement
Use logarithmic differentiation to find dy/dx for y=(1+x)^(1/x).
Homework Equations
dy/dx
The Attempt at a Solution
ln y = ln (1+x)^(1/x)
= (1/x) ln (1+x)
(dy/dx) /y = (-1/x^2)(ln(1+x) + (1/x)(1/(1+x)
dy/dx = y [(-1/x^2) ( ln(1+x) + x/(1+x) ]
or (-1/x^2)( ln(1+x) + x/(1+x) ) (1+x)^(1/x)