Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 x e^{y} - 5 \sin{\left(x \right)} \cos{\left(y^{3} \right)}=-19

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