## Calculus (3rd Edition)

$$f'(x)= \frac{(x+1)(\frac{7}{2}x^{5/2}+\frac{3}{2}x^{1/2})-(x^{7/2}+x^{3/2})}{(x+1)^2}.$$
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 $$f'(x)= \frac{(x+1)(\frac{7}{2}x^{5/2}+\frac{3}{2}x^{1/2})-(x^{7/2}+x^{3/2})}{(x+1)^2}.$$