Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 \sqrt{x} \sqrt{y} - 7 e^{x} e^{y}=43

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{4 \sqrt{x} y'}{\sqrt{y}} - 7 y' e^{x} e^{y} - 7 e^{x} e^{y} - \frac{4 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{7 \sqrt{x} \sqrt{y} e^{x + y} + 4 y}{7 \sqrt{x} \sqrt{y} e^{x + y} + 4 x}