Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle \sqrt{3} \sqrt{y} e^{x^{2}} - 6 \log{\left(y \right)} \cos{\left(x^{3} \right)}=-43$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 18 x^{2} \log{\left(y \right)} \sin{\left(x^{3} \right)} + 2 \sqrt{3} x \sqrt{y} e^{x^{2}} - \frac{6 y' \cos{\left(x^{3} \right)}}{y} + \frac{\sqrt{3} y' e^{x^{2}}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 x \left(9 x y^{\frac{3}{2}} \log{\left(y \right)} \sin{\left(x^{3} \right)} + \sqrt{3} y^{2} e^{x^{2}}\right)}{12 \sqrt{y} \cos{\left(x^{3} \right)} - \sqrt{3} y e^{x^{2}}}$