How do you use logarithmic differentiation to find the derivative of ##y=(tanx)^(1/x)##?

##y=(tanx)^(1/x)##

##lny=ln((tanx)^(1/x))##

##lny=1/xln(tanx)## Differentiate implicitly.

##1/y (dy)/(dx)=-1/x^2ln(tan(x))+1/x(1/tan(x) sec^2(x))##

At this point it's nice to simplify using ##1/tanx=cotx## and ##cotx * secx=cscx##

So, ##1/y (dy)/(dx)=-1/x^2ln(tan(x))+1/x(cscx secx)##

## (dy)/(dx)=y(-1/x^2ln(tan(x))+1/x(cscx secx))##

## (dy)/(dx) =( tanx)^(1/x)(-1/x^2ln(tan(x))+1/x(cscx secx))## ##" "## (Not pretty, but correct.)

## (dy)/(dx) =( tanx)^(1/x)(x(cscx secx)-ln(tan(x)))*1/x^2## ##" "## (Isn't a whole lot better.)