Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 y^{3} \log{\left(x \right)} - 5 e^{x^{3}} \sin{\left(y^{2} \right)}=42

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 15 x^{2} e^{x^{3}} \sin{\left(y^{2} \right)} + 12 y^{2} y' \log{\left(x \right)} - 10 y y' e^{x^{3}} \cos{\left(y^{2} \right)} + \frac{4 y^{3}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{15 x^{3} e^{x^{3}} \sin{\left(y^{2} \right)} - 4 y^{3}}{2 x y \left(6 y \log{\left(x \right)} - 5 e^{x^{3}} \cos{\left(y^{2} \right)}\right)}