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Verified Answer :

f(x) = x^1/2 - x^-1/2
f'(x) = 1/2x^-1/2 - (-1/2x^-3/2)
f'(x) = 1/2sqrt(x) + 1/2(sqrt(x))^3
f'(x) = x + 1/2(sqrt(x))^3

remember that x^-m=1/(x^m)
first, conver to exponential
\(\sqrt{x} - \frac{1}{ \sqrt{x} } =x^ \frac{1}{2}-x^ \frac{-1}{2}\)
now do the thingummy
(1/2)(x^(-1/2))-(-1/2)(x^(-3/2))
convert to fraction form
\(\frac{1}{2 \sqrt{x} } + \frac{1}{2 \sqrt{x^\frac{3}{2}} }\)
\(\frac{1}{2 \sqrt{x} } + \frac{1}{2x \sqrt{x} }\)
dat is the answer