Here is one way.
f(x) = x^(sqrt(x))
Let y = f(x)
So,
y = x^(sqrt(x))
Take the ln of both sides,
ln(y) = sqrt(x) *ln(x)
Differentiate both sides with respect to x,
(1/y)y' = [sqrt(x) *1/x] +[ln(x) *(1/2)x^(-1/2)]
(1/y)y' = [1/sqrt(x)] +[1/(2sqrt(x)) *ln(x)]
Factor the RHS by 1/sqrt(x),
(1/y)y' = (1/sqrt(x))[1 +(1/2)ln(x)]
Multiply both sides by y,
And since 1/sqrt(x) = x^(-1/2),
y' = (x^sqrt(x))*(x^(-1/2))[1 +(1/2)ln(x)]
y' = [x^(sqrt(x) -(1/2)][1 +(1/2)ln(x)] -----answer at the back of the book.