Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 x \cos{\left(y \right)} + \cos{\left(x \right)}=-43$

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