Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 4 x \sin{\left(y^{2} \right)} + \sqrt{6} \sqrt{y} \cos{\left(x^{3} \right)}=-6$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 3 \sqrt{6} x^{2} \sqrt{y} \sin{\left(x^{3} \right)} - 8 x y y' \cos{\left(y^{2} \right)} - 4 \sin{\left(y^{2} \right)} + \frac{\sqrt{6} y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{6 \sqrt{6} x^{2} y \sin{\left(x^{3} \right)} + 8 \sqrt{y} \sin{\left(y^{2} \right)}}{16 x y^{\frac{3}{2}} \cos{\left(y^{2} \right)} - \sqrt{6} \cos{\left(x^{3} \right)}}$