Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle \sqrt{5} \sqrt{x} e^{y^{2}} - 5 \sqrt{5} \sqrt{y} e^{x^{2}}=-43$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 2 \sqrt{5} \sqrt{x} y y' e^{y^{2}} - 10 \sqrt{5} x \sqrt{y} e^{x^{2}} - \frac{5 \sqrt{5} y' e^{x^{2}}}{2 \sqrt{y}} + \frac{\sqrt{5} e^{y^{2}}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 20 x^{\frac{3}{2}} y e^{x^{2}} + \sqrt{y} e^{y^{2}}}{5 \sqrt{x} e^{x^{2}} - 4 x y^{\frac{3}{2}} e^{y^{2}}}$