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

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