Answer
$$ f'(x)= \frac{(x+1)(\frac{7}{2}x^{5/2}+\frac{3}{2}x^{1/2})-(x^{7/2}+x^{3/2})}{(x+1)^2}.$$
Work Step by Step
We rewrite $ f(x)$ as follows
$$ f(x)=\frac{x^{3/2}(x^2+1)}{x+1}=\frac{x^{7/2}+x^{3/2}}{x+1}.$$
Now, the derivative $ f'(x)$, using the quotient rule, is given by
$$ f'(x)= \frac{(x+1)(\frac{7}{2}x^{5/2}+\frac{3}{2}x^{1/2})-(x^{7/2}+x^{3/2})}{(x+1)^2}.$$
