Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 \sqrt{2} \sqrt{y} \log{\left(x \right)} + \cos{\left(x \right)} \cos{\left(y \right)}=29

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