Answer
\[\frac{{dy}}{{dx}} = \frac{{13y - 18{x^2}}}{{21{y^2} - 13x}}\]
Work Step by Step
\[\begin{gathered}
6{x^3} + 7{y^3} = 13xy \hfill \\
\hfill \\
implicit\,\,differentiation \hfill \\
\hfill \\
\frac{d}{{dx}}\,\,\left[ {6{x^3} + 7{y^3}} \right] = \frac{d}{{dx}}\,\,\left[ {13xy} \right] \hfill \\
\hfill \\
use\,\,product\,\,rule \hfill \\
\hfill \\
18{x^2} + 21{y^2}y' = 13xy' + 13y \hfill \\
\hfill \\
collect\,\,like\,\,terms \hfill \\
\hfill \\
21{y^2}y' - 13xy' = 13y - 18{x^2} \hfill \\
\hfill \\
solve\,\,for\,\,{y^,} \hfill \\
\hfill \\
y'\,\left( {21{y^2} - 13x} \right) = 13y - 18{x^2} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{13y - 18{x^2}}}{{21{y^2} - 13x}} \hfill \\
\hfill \\
\end{gathered} \]
