Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 5 \sqrt{5} \sqrt{x} e^{y} + 9 e^{x} \cos{\left(y^{2} \right)}=34

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 5 \sqrt{5} \sqrt{x} y' e^{y} - 18 y y' e^{x} \sin{\left(y^{2} \right)} + 9 e^{x} \cos{\left(y^{2} \right)} + \frac{5 \sqrt{5} e^{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{18 \sqrt{x} e^{x} \cos{\left(y^{2} \right)} + 5 \sqrt{5} e^{y}}{2 \left(18 \sqrt{x} y e^{x} \sin{\left(y^{2} \right)} - 5 \sqrt{5} x e^{y}\right)}