Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 14 \sqrt{2} \sqrt{y} e^{x^{2}} + 9 y^{2} \sin{\left(x^{2} \right)}=-21$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 28 \sqrt{2} x \sqrt{y} e^{x^{2}} + 18 x y^{2} \cos{\left(x^{2} \right)} + 18 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} y' e^{x^{2}}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 x \left(9 y^{\frac{5}{2}} \cos{\left(x^{2} \right)} + 14 \sqrt{2} y e^{x^{2}}\right)}{18 y^{\frac{3}{2}} \sin{\left(x^{2} \right)} + 7 \sqrt{2} e^{x^{2}}}$