Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 e^{x^{3}} \cos{\left(y^{3} \right)} - 6 \log{\left(x \right)} \log{\left(y \right)}=16

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