Find the derivative of f(x) = (x + 1) (cos x)

Answer

To find the derivative of the function \( f(x) = (x + 1) \cos x \), we will use the product rule of differentiation. The product rule states that if you have a function that is the product of two functions, say \( u \) and \( v \), then the derivative of the product \( uv \) is given by:

\[ (uv)' = u'v + uv' \]

### Step-by-Step Solution:

1. **Identify the Functions**:
Let \( u = x + 1 \) and \( v = \cos x \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ u' = \frac{d}{dx}(x + 1) = 1 \]
- The derivative of \( v \) is:
\[ v' = \frac{d}{dx}(\cos x) = -\sin x \]

3. **Apply the Product Rule**:
Using the product rule:
\[ f'(x) = u'v + uv' \]
Substitute \( u \), \( u' \), \( v \), and \( v' \):
\[ f'(x) = (1)(\cos x) + (x + 1)(-\sin x) \]

4. **Simplify the Expression**:
Now simplify the expression:
\[ f'(x) = \cos x - (x + 1)\sin x \]
This can be expanded to:
\[ f'(x) = \cos x - x \sin x - \sin x \]

5. **Final Result**:
Therefore, the derivative of \( f(x) \) is:
\[ f'(x) = \cos x - x \sin x - \sin x \]

### Summary:
The derivative of the function \( f(x) = (x + 1) \cos x \) is:
\[ f'(x) = \cos x - x \sin x - \sin x \]