From the rewrite in the previous step we can see that,

\[F\left( {x,y} \right) = {x^2}{y^4} - 3 - \sin \left( {xy} \right)\]

We can now simply use the formula we derived in the notes to get the derivative.

\[\frac{{dy}}{{dx}} = - \frac{{{F_x}}}{{{F_y}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{2x{y^4} - y\cos \left( {xy} \right)}}{{4{x^2}{y^3} - x\cos \left( {xy} \right)}} = \frac{{y\cos \left( {xy} \right) - 2x{y^4}}}{{4{x^2}{y^3} - x\cos \left( {xy} \right)}}}}\]

Note that in for the second form of the answer we simply moved the “-” in front of the fraction up to the numerator and multiplied it through. We could just have easily done this with the denominator instead if we’d wanted to.