Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 y^{3} \log{\left(x \right)} - 4 e^{y^{2}} \cos{\left(x^{2} \right)}=45

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 8 x e^{y^{2}} \sin{\left(x^{2} \right)} + 9 y^{2} y' \log{\left(x \right)} - 8 y y' e^{y^{2}} \cos{\left(x^{2} \right)} + \frac{3 y^{3}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{2} e^{y^{2}} \sin{\left(x^{2} \right)} + 3 y^{3}}{x y \left(- 9 y \log{\left(x \right)} + 8 e^{y^{2}} \cos{\left(x^{2} \right)}\right)}