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} \cos{\left(y \right)} - 7 y^{2} e^{x^{2}}=-7

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 5 \sqrt{5} \sqrt{x} y' \sin{\left(y \right)} - 14 x y^{2} e^{x^{2}} - 14 y y' e^{x^{2}} - \frac{5 \sqrt{5} \cos{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{28 x^{\frac{3}{2}} y^{2} e^{x^{2}} + 5 \sqrt{5} \cos{\left(y \right)}}{2 \left(- 14 \sqrt{x} y e^{x^{2}} + 5 \sqrt{5} x \sin{\left(y \right)}\right)}