Find `dy/dx if 2x-y=sinx`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x - y = \sin x\), we will follow these steps:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 2x - y = \sin x \]
Now, we will differentiate both sides with respect to \(x\).

### Step 2: Apply differentiation
Differentiating the left side:
- The derivative of \(2x\) is \(2\).
- The derivative of \(-y\) is \(-\frac{dy}{dx}\) (using the chain rule).

Differentiating the right side:
- The derivative of \(\sin x\) is \(\cos x\).

Putting it all together, we have:
\[ \frac{d}{dx}(2x) - \frac{d}{dx}(y) = \frac{d}{dx}(\sin x) \]
This simplifies to:
\[ 2 - \frac{dy}{dx} = \cos x \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we will isolate \(\frac{dy}{dx}\):
\[ -\frac{dy}{dx} = \cos x - 2 \]
Multiplying both sides by \(-1\) gives:
\[ \frac{dy}{dx} = 2 - \cos x \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 2 - \cos x \]

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