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{y} e^{x} + 2 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=13

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 6 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + 6 \sqrt{7} \sqrt{y} e^{x} + 6 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + \frac{3 \sqrt{7} y' e^{x}}{\sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{2 x^{2} \sqrt{y} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + 2 \sqrt{7} y e^{x}}{2 y^{\frac{5}{2}} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + \sqrt{7} e^{x}}