Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{2} \sqrt{x} \log{\left(y \right)} + 4 y^{3} \cos{\left(x^{3} \right)}=3

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{6 \sqrt{2} \sqrt{x} y'}{y} - 12 x^{2} y^{3} \sin{\left(x^{3} \right)} + 12 y^{2} y' \cos{\left(x^{3} \right)} + \frac{3 \sqrt{2} \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(4 x^{\frac{5}{2}} y^{3} \sin{\left(x^{3} \right)} - \sqrt{2} \log{\left(y \right)}\right)}{2 \left(2 \sqrt{x} y^{3} \cos{\left(x^{3} \right)} + \sqrt{2} x\right)}