Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 2 \sqrt{7} x^{2} \sqrt{y} + 2 y^{2} \cos{\left(x \right)}=34$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{\sqrt{7} x^{2} y'}{\sqrt{y}} - 4 \sqrt{7} x \sqrt{y} - 2 y^{2} \sin{\left(x \right)} + 4 y y' \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{4 \sqrt{7} x y + 2 y^{\frac{5}{2}} \sin{\left(x \right)}}{\sqrt{7} x^{2} - 4 y^{\frac{3}{2}} \cos{\left(x \right)}}$