Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle x \sin{\left(y^{2} \right)} - 18 \sqrt{y} \sin{\left(x^{2} \right)}=-23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 36 x \sqrt{y} \cos{\left(x^{2} \right)} + 2 x y y' \cos{\left(y^{2} \right)} + \sin{\left(y^{2} \right)} - \frac{9 y' \sin{\left(x^{2} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{36 x y \cos{\left(x^{2} \right)} - \sqrt{y} \sin{\left(y^{2} \right)}}{2 x y^{\frac{3}{2}} \cos{\left(y^{2} \right)} - 9 \sin{\left(x^{2} \right)}}$