Prove that f(x)=2square rootx(lnx-2) is the antiderivative of f(x)=lnx/square rootx , x>0

To prove that the function F(x) = 2(sqrt x)(lnx-2) is the
antiderivative of f(x) = lnx/sqrt x, we'll have to differentiate
F(x).

F'(x) = f(x)

We'll
differentiate F(x) using product rule:

F'(x) = 2(sqrt
x)'*(lnx-2) + 2(sqrt x)*(lnx-2)'

F'(x) = (lnx-2)/sqrtx +
(2sqrt x)/x

We'll multiply the 1st term by
sqrtx:

F'(x) = (sqrtx)(lnx-2)/x + (2sqrt
x)/x

F'(x) = (sqrtx)(lnx - 2 +
2)/x

F'(x) =
(sqrtx)(lnx)/x

F'(x) = (lnx)/sqrt x =
f(x)

We notice that differentiating F(x),
we've get f(x). Therefore F'(x) = f(x): [2(sqrt x)(lnx-2)]' = ln x/sqrt
x.