Question

What is the derivative of `x^((1/5)(lnx))`?

Answer 1

`2/5*x^((1/5*ln x-1))* ln x`

Explanation:

Let `y = x^((1/5*ln x))` then take logarithm of both sides of the equation

`ln y = ln x^((1/5*ln x))`

`ln y = (1/5 ln x )*( ln x)`

`ln y = 1/5*(ln x)^2`

using implicit differentiation

`1/y * y' = 1/5* 2* ln x*1/x`

then solve for `y'`

`y' = (2y)/(5x) ln x`

replace `y` with the equivalent `x^((1/5*ln x))`

`y' = (2x^((1/5*ln x)))/(5x) ln x`

then simplification

`y' = 2/5*x^((1/5*ln x-1))* ln x`