Answer
\[{F^,}\,\left( x \right) = - 19{x^{ - 2}} - 4{x^{ - 1/2}}\]
Work Step by Step
\[\begin{gathered}
f\,\left( x \right) = 19{x^{ - 1}} - 8\sqrt x \hfill \\
Write\,\sqrt x \,\,as\,\,{x^{1/2}} \hfill \\
Find\,\,the\,\,derivative\, \hfill \\
{F^,}\,\left( x \right) = 19{x^{ - 1}} - 8{x^{1/2}} \hfill \\
find\,\,the\,\,derivative\, \hfill \\
{F^,}\,\left( x \right) = \frac{d}{{dx}}\,\,\left[ {19{x^{ - 1}} - 8{x^{1/2}}} \right] \hfill \\
Use\,\,the\,\,power\,\,rule \hfill \\
\frac{d}{{dx}}\,\,\left[ {{x^n}} \right] = n{x^{n - 1}} \hfill \\
Then \hfill \\
{F^,}\,\left( x \right) = 19\,\left( { - 1} \right){x^{ - 1 - 1}} - 8\,\left( {\frac{1}{2}} \right){x^{1/2 - 1}} \hfill \\
{F^,}\,\left( x \right) = - 19{x^{ - 2}} - 4{x^{ - 1/2}} \hfill \\
\hfill \\
\end{gathered} \]
