Answer
\[\frac{{dy}}{{dx}} = \frac{{18x - 21}}{{2\sqrt {9{x^2} - 21x + 16} }}\]
Work Step by Step
\[\begin{gathered}
y = \sqrt {\,{{\left( {3x - 4} \right)}^2} + 3x} \hfill \\
\hfill \\
rewrite \hfill \\
\hfill \\
y = {\left( {\,{{\left( {3x - 4} \right)}^2} + 3x} \right)^{1/2}} \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,2\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
\,\,y = {u^n} \to \frac{{dy}}{{dx}} = n{u^{n - 1}}{u^,} \hfill \\
\hfill \\
Therefore \hfill \\
\hfill \\
y = \frac{1}{2}{\left( {\,{{\left( {3x - 4} \right)}^2} + 3x} \right)^{ - 1/2}}\frac{d}{{dx}}\left[ {\,{{\left( {3x - 4} \right)}^2} + 3x} \right] \hfill \\
\hfill \\
differentiate \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{2\,\left( {3x - 4} \right)\,\left( 3 \right) + 3}}{{2\sqrt {\,{{\left( {3x - 4} \right)}^2} + 3x} }} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{18x - 24 + 3}}{{2\sqrt {9{x^2} - 24x + 16 + 3x} }} \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{18x - 21}}{{2\sqrt {9{x^2} - 21x + 16} }} \hfill \\
\hfill \\
\end{gathered} \]
