Find `dy/dx if sin2x-2y=cosx`

Answer

To find \(\frac{dy}{dx}\) for the equation \( \sin(2x) - 2y = \cos(x) \), we will differentiate both sides with respect to \(x\).

### Step-by-step Solution:

1. **Differentiate both sides of the equation:**
\[ \frac{d}{dx}(\sin(2x) - 2y) = \frac{d}{dx}(\cos(x)) \]

2. **Apply the differentiation rules:**
- The derivative of \(\sin(2x)\) is \(2\cos(2x)\) (using the chain rule).
- The derivative of \(-2y\) is \(-2\frac{dy}{dx}\) (using the constant multiple rule).
- The derivative of \(\cos(x)\) is \(-\sin(x)\).

Therefore, we have:
\[ 2\cos(2x) - 2\frac{dy}{dx} = -\sin(x) \]

3. **Rearranging the equation:**
To isolate \(\frac{dy}{dx}\), we will move \(-2\frac{dy}{dx}\) to the right side:
\[ 2\cos(2x) + \sin(x) = 2\frac{dy}{dx} \]

4. **Solve for \(\frac{dy}{dx}\):**
Divide both sides by 2:
\[ \frac{dy}{dx} = \frac{2\cos(2x) + \sin(x)}{2} \]

5. **Final expression:**
Simplifying gives us:
\[ \frac{dy}{dx} = \cos(2x) + \frac{1}{2}\sin(x) \]

### Final Answer:
\[ \frac{dy}{dx} = \cos(2x) + \frac{1}{2}\sin(x) \]