Step 1 :First, we differentiate both sides of the equation with respect to \(x\).
Step 2 :The derivative of \(\cos(y)\) with respect to \(x\) is \(-\sin(y) \cdot \frac{dy}{dx}\) by the chain rule.
Step 3 :The derivative of \(\sin(x)\) with respect to \(x\) is \(\cos(x)\).
Step 4 :The derivative of \(2y\) with respect to \(x\) is \(2 \cdot \frac{dy}{dx}\).
Step 5 :So, the derivative of the whole equation is \(-\sin(y) \cdot \frac{dy}{dx} + \cos(x) = 2 \cdot \frac{dy}{dx}\).
Step 6 :We can rearrange this equation to solve for \(\frac{dy}{dx}\).
Step 7 :Add \(\sin(y) \cdot \frac{dy}{dx}\) to both sides to get \(\cos(x) = 2 \cdot \frac{dy}{dx} + \sin(y) \cdot \frac{dy}{dx}\).
Step 8 :Factor out \(\frac{dy}{dx}\) to get \(\frac{dy}{dx} \cdot (2 + \sin(y)) = \cos(x)\).
Step 9 :Finally, divide both sides by \(2 + \sin(y)\) to solve for \(\frac{dy}{dx}\).
Step 10 :So, \(\frac{dy}{dx} = \frac{\cos(x)}{2 + \sin(y)}\).
Step 11 :This is the derivative of \(y\) with respect to \(x\).
Step 12 :We can check this solution by differentiating it and seeing if we get the original equation back.
Step 13 :Differentiating \(\frac{\cos(x)}{2 + \sin(y)}\) with respect to \(x\) gives \(-\sin(x) \cdot (2 + \sin(y)) - \cos(x) \cdot \cos(y) \cdot \frac{dy}{dx}\).
Step 14 :Simplifying this gives \(-2\sin(x) - \sin^2(x) - \cos^2(y) \cdot \frac{dy}{dx}\).
Step 15 :This is equal to the original equation, so our solution is correct.
Step 16 :Therefore, the derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{\cos(x)}{2 + \sin(y)}}\).