Answer
$f'\left( x \right) = \frac{3}{2}{x^{1/2}} - 3{x^{ - 4}}$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^{3/2}} + {x^{ - 3}} \cr
& {\text{Differentiate the function}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^{3/2}} + {x^{ - 3}}} \right] \cr
& {\text{Use the sum diffence rules for differentiation }} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^{3/2}}} \right] + \frac{d}{{dx}}\left[ {{x^{ - 3}}} \right] \cr
& {\text{Apply the power rule: }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr
& f'\left( x \right) = \frac{3}{2}{x^{3/2 - 1}} - 3{x^{ - 3 - 1}} \cr
& {\text{Simplify}} \cr
& f'\left( x \right) = \frac{3}{2}{x^{1/2}} - 3{x^{ - 4}} \cr} $$
