Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 \sqrt{y} \sin{\left(x \right)} - 6 \cos{\left(x \right)} \cos{\left(y \right)}=-19

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 7 \sqrt{y} \cos{\left(x \right)} + 6 y' \sin{\left(y \right)} \cos{\left(x \right)} + 6 \sin{\left(x \right)} \cos{\left(y \right)} - \frac{7 y' \sin{\left(x \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 6 \sqrt{y} \sin{\left(x \right)} \cos{\left(y \right)} + 7 y \cos{\left(x \right)}\right)}{12 \sqrt{y} \sin{\left(y \right)} \cos{\left(x \right)} - 7 \sin{\left(x \right)}}