Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 \sqrt{x} \cos{\left(y^{3} \right)} - 9 e^{y^{2}} \cos{\left(x^{3} \right)}=10

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 24 \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} + 27 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} - 18 y y' e^{y^{2}} \cos{\left(x^{3} \right)} - \frac{4 \cos{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{27 x^{\frac{5}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} - 4 \cos{\left(y^{3} \right)}}{6 y \left(3 \sqrt{x} e^{y^{2}} \cos{\left(x^{3} \right)} - 4 x y \sin{\left(y^{3} \right)}\right)}