Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 4 \sqrt{2} \sqrt{y} \sin{\left(x^{3} \right)} - 6 e^{x^{2}} \cos{\left(y \right)}=49$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 12 \sqrt{2} x^{2} \sqrt{y} \cos{\left(x^{3} \right)} - 12 x e^{x^{2}} \cos{\left(y \right)} + 6 y' e^{x^{2}} \sin{\left(y \right)} - \frac{2 \sqrt{2} y' \sin{\left(x^{3} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{6 x \left(\sqrt{2} x y \cos{\left(x^{3} \right)} + \sqrt{y} e^{x^{2}} \cos{\left(y \right)}\right)}{3 \sqrt{y} e^{x^{2}} \sin{\left(y \right)} - \sqrt{2} \sin{\left(x^{3} \right)}}$