Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 x \sqrt{y} - 8 e^{x^{2}} \cos{\left(y \right)}=-46$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 16 x e^{x^{2}} \cos{\left(y \right)} + \frac{3 x y'}{2 \sqrt{y}} + 3 \sqrt{y} + 8 y' e^{x^{2}} \sin{\left(y \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 \left(16 x \sqrt{y} e^{x^{2}} \cos{\left(y \right)} - 3 y\right)}{3 x + 16 \sqrt{y} e^{x^{2}} \sin{\left(y \right)}}$