$Q$ is the point $(x,\sqrt[\phantom{2}]{x}+3),$ find the slope of the secant line $PQ$ for the following values of $x.$

Enter at least $3$ decimal places in your answers.

If $x=9\mathrm{.}1,$ the slope of $PQ$ is: $0\mathrm{.}16620625$

If $x=9\mathrm{.}01,$ the slope of $PQ$ is: $0\mathrm{.}16662039$

If $x=8\mathrm{.}9,$ the slope of $PQ$ is: $0\mathrm{.}16713221$

If $x=8\mathrm{.}99,$ the slope of $PQ$ is: $0\mathrm{.}16671298$

Based on the above results, guess the instantaneous rate of change of $y=\sqrt[\phantom{2}]{x}+3$ at $P(9,6).$