Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 x^{3} \cos{\left(y \right)} - 3 e^{y^{2}} \log{\left(x \right)}=-16

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 9 x^{3} y' \sin{\left(y \right)} + 27 x^{2} \cos{\left(y \right)} - 6 y y' e^{y^{2}} \log{\left(x \right)} - \frac{3 e^{y^{2}}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{9 x^{3} \cos{\left(y \right)} - e^{y^{2}}}{x \left(3 x^{3} \sin{\left(y \right)} + 2 y e^{y^{2}} \log{\left(x \right)}\right)}