Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 \sqrt{x} \cos{\left(y \right)} + \cos{\left(x \right)}=-1$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 8 \sqrt{x} y' \sin{\left(y \right)} - \sin{\left(x \right)} - \frac{4 \cos{\left(y \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 \sqrt{x} \cos{\left(y \right)} + x \sin{\left(x \right)}}{8 x^{\frac{3}{2}} \sin{\left(y \right)}}$