Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 \sqrt{x} e^{y^{2}} + 3 y e^{x^{2}}=36$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 6 \sqrt{x} y y' e^{y^{2}} + 6 x y e^{x^{2}} + 3 y' e^{x^{2}} + \frac{3 e^{y^{2}}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{4 x^{\frac{3}{2}} y e^{x^{2}} + e^{y^{2}}}{2 \sqrt{x} e^{x^{2}} + 4 x y e^{y^{2}}}$