Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 5 \sqrt{7} \sqrt{x} y - 2 e^{x} \sin{\left(y \right)}=8

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