Problem: y=sqrtx ^1/x. Find DY/DX.

Question

ln(f(x))=lnx^1/2x or lnx to the 1/2x

1/2x lnx

product rule

(1/4x^2)(lnx)+(1/x)(1/2x)
is the product rule correct?
What do i do after this product rule?

Answer 1

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