Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 y \log{\left(x \right)} - 8 \log{\left(y \right)} \sin{\left(x^{3} \right)}=18

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 24 x^{2} \log{\left(y \right)} \cos{\left(x^{3} \right)} - 3 y' \log{\left(x \right)} - \frac{8 y' \sin{\left(x^{3} \right)}}{y} - \frac{3 y}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{3 y \left(8 x^{3} \log{\left(y \right)} \cos{\left(x^{3} \right)} + y\right)}{x \left(3 y \log{\left(x \right)} + 8 \sin{\left(x^{3} \right)}\right)}