Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 \sqrt{5} \sqrt{x} \log{\left(y \right)} + 2 \cos{\left(x \right)} \cos{\left(y \right)}=1

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{6 \sqrt{5} \sqrt{x} y'}{y} - 2 y' \sin{\left(y \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(y \right)} - \frac{3 \sqrt{5} \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{y \left(2 \sqrt{x} \sin{\left(x \right)} \cos{\left(y \right)} + 3 \sqrt{5} \log{\left(y \right)}\right)}{2 \sqrt{x} y \sin{\left(y \right)} \cos{\left(x \right)} + 6 \sqrt{5} x}