Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{3} \sqrt{x} e^{y} - 5 \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)}=-46

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \sqrt{3} \sqrt{x} y' e^{y} + 10 x \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 10 y y' \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} - \frac{\sqrt{3} e^{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 20 x^{\frac{3}{2}} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + \sqrt{3} e^{y}}{2 \left(10 \sqrt{x} y \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} - \sqrt{3} x e^{y}\right)}