Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{5} \sqrt{x} \cos{\left(y^{3} \right)} - 8 e^{x} \log{\left(y \right)}=6

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 \sqrt{5} \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} - 8 e^{x} \log{\left(y \right)} - \frac{8 y' e^{x}}{y} + \frac{3 \sqrt{5} \cos{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 8 \sqrt{x} e^{x} \log{\left(y \right)} + 3 \sqrt{5} \cos{\left(y^{3} \right)}\right)}{2 \left(4 \sqrt{x} e^{x} + 9 \sqrt{5} x y^{3} \sin{\left(y^{3} \right)}\right)}