Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle \sqrt{6} \sqrt{y} e^{x^{2}} - 8 y^{3} \sin{\left(x \right)}=48$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 2 \sqrt{6} x \sqrt{y} e^{x^{2}} - 8 y^{3} \cos{\left(x \right)} - 24 y^{2} y' \sin{\left(x \right)} + \frac{\sqrt{6} y' e^{x^{2}}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 \left(\sqrt{6} x y e^{x^{2}} - 4 y^{\frac{7}{2}} \cos{\left(x \right)}\right)}{48 y^{\frac{5}{2}} \sin{\left(x \right)} - \sqrt{6} e^{x^{2}}}$