Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 \sqrt{7} \sqrt{x} e^{y^{3}} - 9 e^{x^{3}} \cos{\left(y \right)}=47

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 \sqrt{7} \sqrt{x} y^{2} y' e^{y^{3}} - 27 x^{2} e^{x^{3}} \cos{\left(y \right)} + 9 y' e^{x^{3}} \sin{\left(y \right)} - \frac{3 \sqrt{7} e^{y^{3}}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{9 x^{\frac{5}{2}} e^{x^{3}} \cos{\left(y \right)} + \sqrt{7} e^{y^{3}}}{3 \left(\sqrt{x} e^{x^{3}} \sin{\left(y \right)} - 2 \sqrt{7} x y^{2} e^{y^{3}}\right)}