Question

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

Answer 1

`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.)