Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 y e^{x} - 8 e^{y} \sin{\left(x \right)}=-47$

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