Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 \sqrt{2} \sqrt{x} y^{3} - 4 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=-46

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 6 \sqrt{2} \sqrt{x} y^{2} y' + 8 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} - 12 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - \frac{\sqrt{2} y^{3}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{\frac{3}{2}} \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} - \sqrt{2} y^{3}}{6 y^{2} \left(2 \sqrt{x} \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} + \sqrt{2} x\right)}