Find `dy/dx if 2x+3y=sinx`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x + 3y = \sin x\), we will differentiate both sides of the equation with respect to \(x\).

### Step-by-Step Solution:

1. **Differentiate both sides of the equation:**
\[ \frac{d}{dx}(2x + 3y) = \frac{d}{dx}(\sin x) \]

2. **Apply the differentiation:**
- The derivative of \(2x\) with respect to \(x\) is \(2\).
- The derivative of \(3y\) with respect to \(x\) is \(3\frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\).

So, we have:
\[ 2 + 3\frac{dy}{dx} = \cos x \]

3. **Isolate \(\frac{dy}{dx}\):**
\[ 3\frac{dy}{dx} = \cos x - 2 \]

4. **Solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{\cos x - 2}{3} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{\cos x - 2}{3} \]