Answer
$r'\left( z \right) = - 5{z^{ - 6}} - \frac{1}{2}{z^{ - 1/2}}$
Work Step by Step
$$\eqalign{
& r\left( z \right) = {z^{ - 5}} - {z^{1/2}} \cr
& {\text{Differentiate the function}} \cr
& r'\left( z \right) = \frac{d}{{dz}}\left[ {{z^{ - 5}} - {z^{1/2}}} \right] \cr
& {\text{Use the sum diffence rules for differentiation }} \cr
& r'\left( z \right) = \frac{d}{{dz}}\left[ {{z^{ - 5}}} \right] - \frac{d}{{dz}}\left[ {{z^{1/2}}} \right] \cr
& {\text{Apply the rules: }}\frac{d}{{dt}}\left[ {{z^n}} \right] = n{t^{n - 1}} \cr
& r'\left( z \right) = - 5{z^{ - 6}} - \frac{1}{2}{z^{ - 1/2}} \cr} $$
