Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 e^{x^{3}} \sin{\left(y \right)} - 5 \log{\left(x \right)} \log{\left(y \right)}=18

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