Derivitive of x^(1/x)?

Question

I feel stupid asking this question but what is the derivitive of x^(1/x)

Answer 1

Let x^(1/x)=y
take ln of both sides
1/x lnx=lny
-lnx/x^2+ 1/x*1/x=y'/y

y'= x^(1/x)* ((-lnx)/x^2+(1/x^2))

Answer 2

http://www.calc101.com/webMathematica/derivatives.jsp#topdoit

finds the Derivitive and shows you step by step on how its done

Answer 3

Use logarithmic differentiation to obtain -x^(-(-1+2*x)/x)*(ln(x)-1)

Answer 4

let y = x^ 1/x
log y = log x^ 1/x = 1/x log x
Differentiating bothsides w.r.t x
d/dx(log y ) = d/dx[ 1/x log x]
1/y dy/dx = 1/x d/dx(log x) + log x * d/dx(1/x)
1/y dy/dx = 1/x* 1/x + log x * [-1/(x^2)]
1/y dy/dx = 1/(x^2) - log x / (x^2)
dy/dx = y* [1-log x]/(x^2)
or dy/dx = [x^1/x]*[1-log x]/(x^2)