Question

How do you differentiate `x^(1/x)`?

Answer 1

Use some version of logarithmic differentiation.

Explanation:

`x^(1/x) = e^(1/xlnx) = e^(lnx/x)`

`d/dx(x^(1/x)) = d/dx (e^(lnx/x)) = e^(lnx/x) * d/dx(lnx/x)`

`= x^(1/x) * ((1/x)*(x) - (lnx)(1))/x^2`

`= x^(1/x)((1-lnx)/x^2)`