Use implicit differentiation to find $\frac{d y}{d x}$.
\[
\cos (y)+\sin (x)=2 y
\]

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 \(\boxed{\frac{\cos(x)}{2 + \sin(y)}}\).