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} e^{y^{2}} - 8 \sin{\left(y \right)} \cos{\left(x^{2} \right)}=-10

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 14 \sqrt{3} \sqrt{x} y y' e^{y^{2}} + 16 x \sin{\left(x^{2} \right)} \sin{\left(y \right)} - 8 y' \cos{\left(x^{2} \right)} \cos{\left(y \right)} - \frac{7 \sqrt{3} e^{y^{2}}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{32 x^{\frac{3}{2}} \sin{\left(x^{2} \right)} \sin{\left(y \right)} - 7 \sqrt{3} e^{y^{2}}}{4 \left(4 \sqrt{x} \cos{\left(x^{2} \right)} \cos{\left(y \right)} + 7 \sqrt{3} x y e^{y^{2}}\right)}