Question

How to find dy/dx of the function y=x^(1/x) using logs and implicit differentiation?

Answer 1

y = x ^(1/x)
Taking logs:-

ln y = ln x^(1/x)
ln y = 1/x ln x

differentiating:-

dy/dx * 1/y = = 1/x * 1/x + lnx * - 1/x^2

dy/dx = (1/x^2 - lnx / x^2 ) * y

= [x ^(1/x) ( 1 - ln x) ]/ x^2 answer