Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 \sqrt{7} \sqrt{x} y^{2} - 7 \sqrt{7} \sqrt{y} \cos{\left(x \right)}=19$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 6 \sqrt{7} \sqrt{x} y y' + 7 \sqrt{7} \sqrt{y} \sin{\left(x \right)} - \frac{7 \sqrt{7} y' \cos{\left(x \right)}}{2 \sqrt{y}} - \frac{3 \sqrt{7} y^{2}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{14 \sqrt{x} y \sin{\left(x \right)} - 3 y^{\frac{5}{2}}}{7 \sqrt{x} \cos{\left(x \right)} + 12 x y^{\frac{3}{2}}}$