Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 18 \sqrt{x} \cos{\left(y \right)} + 5 \log{\left(x \right)} \log{\left(y \right)}=-4$

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