Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - e^{x^{3}} e^{y^{3}} + 8 \log{\left(x \right)} \log{\left(y \right)}=42

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