Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 18 \sqrt{x} \log{\left(y \right)} + 6 e^{x^{3}} e^{y^{3}}=23

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{18 \sqrt{x} y'}{y} + 18 x^{2} e^{x^{3}} e^{y^{3}} + 18 y^{2} y' e^{x^{3}} e^{y^{3}} - \frac{9 \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 2 x^{\frac{5}{2}} e^{x^{3} + y^{3}} + \log{\left(y \right)}\right)}{2 \left(\sqrt{x} y^{3} e^{x^{3} + y^{3}} - x\right)}