Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 18 \sqrt{x} y^{2} - 6 \sqrt{y} \sin{\left(x^{2} \right)}=45$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 36 \sqrt{x} y y' - 12 x \sqrt{y} \cos{\left(x^{2} \right)} - \frac{3 y' \sin{\left(x^{2} \right)}}{\sqrt{y}} + \frac{9 y^{2}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 4 x^{\frac{3}{2}} y \cos{\left(x^{2} \right)} + 3 y^{\frac{5}{2}}}{\sqrt{x} \sin{\left(x^{2} \right)} - 12 x y^{\frac{3}{2}}}$