Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 \sqrt{7} x \sqrt{y} + 6 \log{\left(y \right)} \cos{\left(x \right)}=-4$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{4 \sqrt{7} x y'}{\sqrt{y}} - 8 \sqrt{7} \sqrt{y} - 6 \log{\left(y \right)} \sin{\left(x \right)} + \frac{6 y' \cos{\left(x \right)}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{3 y^{\frac{3}{2}} \log{\left(y \right)} \sin{\left(x \right)} + 4 \sqrt{7} y^{2}}{2 \sqrt{7} x y - 3 \sqrt{y} \cos{\left(x \right)}}$