Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{6} \sqrt{x} \cos{\left(y \right)} - 6 \log{\left(x \right)} \sin{\left(y \right)}=44

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