Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 e^{x^{2}} \sin{\left(y \right)} + 8 e^{y^{2}} \log{\left(x \right)}=-43

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 x e^{x^{2}} \sin{\left(y \right)} + 16 y y' e^{y^{2}} \log{\left(x \right)} + 9 y' e^{x^{2}} \cos{\left(y \right)} + \frac{8 e^{y^{2}}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{18 x^{2} e^{x^{2}} \sin{\left(y \right)} + 8 e^{y^{2}}}{x \left(16 y e^{y^{2}} \log{\left(x \right)} + 9 e^{x^{2}} \cos{\left(y \right)}\right)}