Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 \sqrt{6} \sqrt{x} y^{2} + 8 \log{\left(y \right)} \cos{\left(x^{3} \right)}=30

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 4 \sqrt{6} \sqrt{x} y y' - 24 x^{2} \log{\left(y \right)} \sin{\left(x^{3} \right)} + \frac{8 y' \cos{\left(x^{3} \right)}}{y} - \frac{\sqrt{6} y^{2}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(24 x^{\frac{5}{2}} \log{\left(y \right)} \sin{\left(x^{3} \right)} + \sqrt{6} y^{2}\right)}{4 \left(2 \sqrt{x} \cos{\left(x^{3} \right)} - \sqrt{6} x y^{2}\right)}