Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 5 \sqrt{6} \sqrt{x} e^{y^{2}} + 9 \sqrt{6} \sqrt{y} \cos{\left(x^{2} \right)}=12$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 10 \sqrt{6} \sqrt{x} y y' e^{y^{2}} - 18 \sqrt{6} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{9 \sqrt{6} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} - \frac{5 \sqrt{6} e^{y^{2}}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{36 x^{\frac{3}{2}} y \sin{\left(x^{2} \right)} + 5 \sqrt{y} e^{y^{2}}}{9 \sqrt{x} \cos{\left(x^{2} \right)} - 20 x y^{\frac{3}{2}} e^{y^{2}}}$