Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 e^{y^{2}} \cos{\left(x^{2} \right)} + 6 \log{\left(x \right)} \cos{\left(y^{2} \right)}=-10

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 16 x e^{y^{2}} \sin{\left(x^{2} \right)} - 16 y y' e^{y^{2}} \cos{\left(x^{2} \right)} - 12 y y' \log{\left(x \right)} \sin{\left(y^{2} \right)} + \frac{6 \cos{\left(y^{2} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{2} e^{y^{2}} \sin{\left(x^{2} \right)} + 3 \cos{\left(y^{2} \right)}}{2 x y \left(4 e^{y^{2}} \cos{\left(x^{2} \right)} + 3 \log{\left(x \right)} \sin{\left(y^{2} \right)}\right)}