Answer
$$m = \frac{{25}}{6}$$
Work Step by Step
$$\eqalign{
& y = - 2{x^{1/2}} + {x^{3/2}},\,\,\,\,\,\,\,x = 9 \cr
& {\text{find the derivative of the function}} \cr
& \frac{{dy}}{{dx}} = {D_x}\left( { - 2{x^{1/2}} + {x^{3/2}}} \right) \cr
& {\text{use sum rule for derivatives}} \cr
& \frac{{dy}}{{dx}} = {D_x}\left( { - 2{x^{1/2}}} \right) + {D_x}\left( {{x^{3/2}}} \right) \cr
& {\text{use the power rule for derivatives}} \cr
& \frac{{dy}}{{dx}} = - 2\left( {\frac{1}{2}} \right){x^{1/2 - 1}} + \frac{3}{2}{x^{3/2 - 1}} \cr
& \frac{{dy}}{{dx}} = - {x^{ - 1/2}} + \frac{3}{2}{x^{1/2}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{\sqrt x }} + \frac{3}{2}\sqrt x \cr
& {\text{find the slope at }}x = 9{\text{ evaluating the derivative at }}x = 9 \cr
& m = {\left. {\frac{{dy}}{{dx}}} \right|_{x = 9}} = - \frac{1}{{\sqrt 9 }} + \frac{3}{2}\sqrt 9 \cr
& {\text{simplifying}} \cr
& m = - \frac{1}{3} + \frac{9}{2} \cr
& m = \frac{{ - 2 + 27}}{6} \cr
& m = \frac{{25}}{6} \cr} $$
