Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{3} \sqrt{x} y^{2} + 7 x^{2} e^{y}=-7

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 \sqrt{3} \sqrt{x} y y' + 7 x^{2} y' e^{y} + 14 x e^{y} - \frac{\sqrt{3} y^{2}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 28 x^{\frac{3}{2}} e^{y} + \sqrt{3} y^{2}}{2 \left(7 x^{\frac{5}{2}} e^{y} - 2 \sqrt{3} x y\right)}