Answer
$G'\left( r \right) = \frac{3}{2}{r^{ - 1/2}} + \frac{3}{2}{r^{1/2}}$
Work Step by Step
$$\eqalign{
& G\left( r \right) = \frac{{3{r^{3/2}} + {r^{5/2}}}}{r} \cr
& {\text{Rewrite the function}}{\text{, distribute the numerator}} \cr
& G\left( r \right) = \frac{{3{r^{3/2}}}}{r} + \frac{{{r^{5/2}}}}{r} \cr
& {\text{Simplify}} \cr
& G\left( r \right) = 3{r^{1/2}} + {r^{3/2}} \cr
& {\text{Differentiating}} \cr
& G'\left( r \right) = \frac{d}{{dr}}\left[ {3{r^{1/2}} + {r^{3/2}}} \right] \cr
& {\text{Use the sum rule for derivatives}} \cr
& G'\left( r \right) = \frac{d}{{dr}}\left[ {3{r^{1/2}}} \right] + \frac{d}{{dr}}\left[ {{r^{3/2}}} \right] \cr
& G'\left( r \right) = 3\frac{d}{{dr}}\left[ {{r^{1/2}}} \right] + \frac{d}{{dr}}\left[ {{r^{3/2}}} \right] \cr
& {\text{Use the power rule }}\frac{d}{{dr}}\left[ {{r^n}} \right] = n{r^{n - 1}} \cr
& G'\left( r \right) = 3\left( {\frac{1}{2}{r^{1/2 - 1}}} \right) + \frac{3}{2}{r^{3/2 - 1}} \cr
& G'\left( r \right) = \frac{3}{2}{r^{ - 1/2}} + \frac{3}{2}{r^{1/2}} \cr} $$
