Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 9 \sqrt{6} x^{2} \sqrt{y} - 5 e^{y^{2}} \sin{\left(x^{2} \right)}=-29$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{9 \sqrt{6} x^{2} y'}{2 \sqrt{y}} - 18 \sqrt{6} x \sqrt{y} - 10 x e^{y^{2}} \cos{\left(x^{2} \right)} - 10 y y' e^{y^{2}} \sin{\left(x^{2} \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{4 x \left(5 \sqrt{y} e^{y^{2}} \cos{\left(x^{2} \right)} + 9 \sqrt{6} y\right)}{9 \sqrt{6} x^{2} + 20 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{2} \right)}}$