Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 7 \sqrt{3} \sqrt{x} y^{2} + 5 e^{y^{2}} \cos{\left(x \right)}=-8

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 14 \sqrt{3} \sqrt{x} y y' + 10 y y' e^{y^{2}} \cos{\left(x \right)} - 5 e^{y^{2}} \sin{\left(x \right)} + \frac{7 \sqrt{3} y^{2}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{10 \sqrt{x} e^{y^{2}} \sin{\left(x \right)} - 7 \sqrt{3} y^{2}}{4 y \left(5 \sqrt{x} e^{y^{2}} \cos{\left(x \right)} + 7 \sqrt{3} x\right)}