Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 8 y^{3} \cos{\left(x \right)} + \sin{\left(x \right)}=-9$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 8 y^{3} \sin{\left(x \right)} + 24 y^{2} y' \cos{\left(x \right)} + \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{y \tan{\left(x \right)}}{3} - \frac{1}{24 y^{2}}$