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

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