Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 x^{2} \cos{\left(y \right)} - 9 \sqrt{5} \sqrt{y} \sin{\left(x \right)}=20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 8 x^{2} y' \sin{\left(y \right)} - 16 x \cos{\left(y \right)} - 9 \sqrt{5} \sqrt{y} \cos{\left(x \right)} - \frac{9 \sqrt{5} y' \sin{\left(x \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(16 x \sqrt{y} \cos{\left(y \right)} + 9 \sqrt{5} y \cos{\left(x \right)}\right)}{16 x^{2} \sqrt{y} \sin{\left(y \right)} - 9 \sqrt{5} \sin{\left(x \right)}}