Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 2 \sqrt{3} \sqrt{y} \sin{\left(x^{3} \right)} + e^{x^{3}} e^{y^{3}}=37

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