Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle x \log{\left(y \right)} - 16 \sqrt{y} \cos{\left(x \right)}=18$

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