Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 \sqrt{5} \sqrt{y} \sin{\left(x^{2} \right)} + 9 \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)}=-2$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 16 \sqrt{5} x \sqrt{y} \cos{\left(x^{2} \right)} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} + 18 y y' \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - \frac{4 \sqrt{5} y' \sin{\left(x^{2} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{x \left(9 \sqrt{y} \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} + 8 \sqrt{5} y \cos{\left(x^{2} \right)}\right)}{9 y^{\frac{3}{2}} \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 2 \sqrt{5} \sin{\left(x^{2} \right)}}$