Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 7 e^{x} \cos{\left(y^{3} \right)} - 3 e^{y} \log{\left(x \right)}=41

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 21 y^{2} y' e^{x} \sin{\left(y^{3} \right)} - 3 y' e^{y} \log{\left(x \right)} + 7 e^{x} \cos{\left(y^{3} \right)} - \frac{3 e^{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{\frac{7 x e^{x} \cos{\left(y^{3} \right)}}{3} - e^{y}}{x \left(7 y^{2} e^{x} \sin{\left(y^{3} \right)} + e^{y} \log{\left(x \right)}\right)}