Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 6 \sqrt{6} \sqrt{y} \sin{\left(x \right)} + 2 e^{x^{3}} \log{\left(y \right)}=14$

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