Answer
$f'(x) = \dfrac{3(\sqrt{x+1} -1)^{1/2}}{4\sqrt{x+1}}$
Work Step by Step
In order to derivate this function you have to apply the chain rule
Let's make an «u» substitution to make it easier
$u = \sqrt{x+1} -1$
$f(u) = u^{3/2}$
Derivate the function:
$f'(u) = \dfrac{3}{2}u^{1/2}u'$
Now let's find u'
$u' = \dfrac{1}{2\sqrt{x+1}}$
Then undo the substitution, simplify and get the answer:
$f'(x) = (\dfrac{3}{2}(\sqrt{x+1} -1)^{1/2})(\dfrac{1}{2\sqrt{x+1}})$
$f'(x) = \dfrac{3(\sqrt{x+1} -1)^{1/2}}{4\sqrt{x+1}}$
