Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 5 x^{2} y^{2} + e^{x} \sin{\left(y^{2} \right)}=-24

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 10 x^{2} y y' + 10 x y^{2} + 2 y y' e^{x} \cos{\left(y^{2} \right)} + e^{x} \sin{\left(y^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{10 x y^{2} + e^{x} \sin{\left(y^{2} \right)}}{2 y \left(5 x^{2} + e^{x} \cos{\left(y^{2} \right)}\right)}