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} \sin{\left(y^{3} \right)} - 8 e^{y^{2}} \log{\left(x \right)}=-32

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