Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 6 x^{2} y^{2} - 2 \sqrt{7} \sqrt{y} e^{x^{2}}=-34$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 12 x^{2} y y' - 4 \sqrt{7} x \sqrt{y} e^{x^{2}} + 12 x y^{2} - \frac{\sqrt{7} y' e^{x^{2}}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 x \left(- 3 y^{\frac{5}{2}} + \sqrt{7} y e^{x^{2}}\right)}{12 x^{2} y^{\frac{3}{2}} - \sqrt{7} e^{x^{2}}}$