If `2x+3y=sinx,"then "dy/dx`= ____________ .

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 derivative**:
- The derivative of \(2x\) is \(2\).
- The derivative of \(3y\) with respect to \(x\) requires the use of the chain rule, which gives us \(3 \frac{dy}{dx}\).
- The derivative of \(\sin x\) is \(\cos x\).

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

3. **Isolate \(\frac{dy}{dx}\)**:
To isolate \(\frac{dy}{dx}\), we rearrange the equation:
\[ 3\frac{dy}{dx} = \cos x - 2 \]

4. **Divide by 3**:
Finally, we divide both sides by 3 to solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{\cos x - 2}{3} \]

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