Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle x^{3} y^{3} + 8 e^{y^{2}} \log{\left(x \right)}=24$

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