Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 x^{3} \cos{\left(y^{3} \right)} - 4 e^{y} \sin{\left(x \right)}=-40

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