Find `dy/dx` when `2x+3y=cosx`.

Answer

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

### Step-by-Step Solution:

1. **Differentiate both sides with respect to \(x\)**:
\[ \frac{d}{dx}(2x + 3y) = \frac{d}{dx}(\cos x) \]

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

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

3. **Isolate \(\frac{dy}{dx}\)**:
- Subtract \(2\) from both sides:
\[ 3 \frac{dy}{dx} = -\sin x - 2 \]
- Divide both sides by \(3\):
\[ \frac{dy}{dx} = \frac{-\sin x - 2}{3} \]

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