Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 \sqrt{3} \sqrt{x} \cos{\left(y \right)} - 6 y \log{\left(x \right)}=-22

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