Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 e^{x} \sin{\left(y \right)} + 9 \log{\left(y \right)} \cos{\left(x^{3} \right)}=-24

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 27 x^{2} \log{\left(y \right)} \sin{\left(x^{3} \right)} - 4 y' e^{x} \cos{\left(y \right)} - 4 e^{x} \sin{\left(y \right)} + \frac{9 y' \cos{\left(x^{3} \right)}}{y} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(27 x^{2} \log{\left(y \right)} \sin{\left(x^{3} \right)} + 4 e^{x} \sin{\left(y \right)}\right)}{- 4 y e^{x} \cos{\left(y \right)} + 9 \cos{\left(x^{3} \right)}}