Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{7} \sqrt{y} \log{\left(x \right)} + 2 y^{2} \sin{\left(x \right)}=10

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 2 y^{2} \cos{\left(x \right)} + 4 y y' \sin{\left(x \right)} - \frac{\sqrt{7} y' \log{\left(x \right)}}{2 \sqrt{y}} - \frac{\sqrt{7} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 2 x y^{\frac{5}{2}} \cos{\left(x \right)} + \sqrt{7} y\right)}{x \left(8 y^{\frac{3}{2}} \sin{\left(x \right)} - \sqrt{7} \log{\left(x \right)}\right)}