Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 18 \sqrt{y} \cos{\left(x^{2} \right)} + 8 \log{\left(y \right)} \sin{\left(x^{2} \right)}=45$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 36 x \sqrt{y} \sin{\left(x^{2} \right)} + 16 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{8 y' \sin{\left(x^{2} \right)}}{y} - \frac{9 y' \cos{\left(x^{2} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 x \left(4 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)} + 9 y^{2} \sin{\left(x^{2} \right)}\right)}{- 8 \sqrt{y} \sin{\left(x^{2} \right)} + 9 y \cos{\left(x^{2} \right)}}$