Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle \sqrt{5} \sqrt{y} \log{\left(x \right)} - 9 e^{y^{3}} \sin{\left(x^{3} \right)}=-11

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 27 x^{2} e^{y^{3}} \cos{\left(x^{3} \right)} - 27 y^{2} y' e^{y^{3}} \sin{\left(x^{3} \right)} + \frac{\sqrt{5} y' \log{\left(x \right)}}{2 \sqrt{y}} + \frac{\sqrt{5} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 27 x^{3} \sqrt{y} e^{y^{3}} \cos{\left(x^{3} \right)} + \sqrt{5} y\right)}{x \left(54 y^{\frac{5}{2}} e^{y^{3}} \sin{\left(x^{3} \right)} - \sqrt{5} \log{\left(x \right)}\right)}