Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 \sqrt{6} \sqrt{x} \sin{\left(y \right)} + 5 y^{2} \sin{\left(x \right)}=7

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 9 \sqrt{6} \sqrt{x} y' \cos{\left(y \right)} + 5 y^{2} \cos{\left(x \right)} + 10 y y' \sin{\left(x \right)} - \frac{9 \sqrt{6} \sin{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 10 \sqrt{x} y^{2} \cos{\left(x \right)} + 9 \sqrt{6} \sin{\left(y \right)}}{2 \left(10 \sqrt{x} y \sin{\left(x \right)} - 9 \sqrt{6} x \cos{\left(y \right)}\right)}