Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle x + 5 \sqrt{y} \cos{\left(x \right)}=-27

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 5 \sqrt{y} \sin{\left(x \right)} + 1 + \frac{5 y' \cos{\left(x \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{2 \sqrt{y}}{5 \cos{\left(x \right)}} + 2 y \tan{\left(x \right)}