Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 8 e^{x^{2}} \log{\left(y \right)} + 2 \sin{\left(x^{3} \right)} \cos{\left(y \right)}=14$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 6 x^{2} \cos{\left(x^{3} \right)} \cos{\left(y \right)} + 16 x e^{x^{2}} \log{\left(y \right)} - 2 y' \sin{\left(x^{3} \right)} \sin{\left(y \right)} + \frac{8 y' e^{x^{2}}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{x y \left(3 x \cos{\left(x^{3} \right)} \cos{\left(y \right)} + 8 e^{x^{2}} \log{\left(y \right)}\right)}{y \sin{\left(x^{3} \right)} \sin{\left(y \right)} - 4 e^{x^{2}}}$