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

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 6 \sqrt{x} y' \cos{\left(y \right)} - \sin{\left(x \right)} - \frac{3 \sin{\left(y \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{\tan{\left(y \right)}}{2 x} - \frac{\sin{\left(x \right)}}{6 \sqrt{x} \cos{\left(y \right)}}$