Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 x \cos{\left(y \right)} + 8 y \cos{\left(x \right)}=-32$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 6 x y' \sin{\left(y \right)} - 8 y \sin{\left(x \right)} + 8 y' \cos{\left(x \right)} - 6 \cos{\left(y \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 y \sin{\left(x \right)} + 3 \cos{\left(y \right)}}{3 x \sin{\left(y \right)} + 4 \cos{\left(x \right)}}$