The derivative of f(X) = x cos x + tan x is

Options

A. x sin x + cos x - `x^(2)`
B. x sin x + cos x + `sec^(2) x`
C. `-x` sin x + cos x + sec x
D. `-x` sin x + cos x + `sec^(2) x`

Answer

To find the derivative of the function \( f(x) = x \cos x + \tan x \), we will use the rules of differentiation, including the product rule and the derivative of standard functions.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = x \cos x + \tan x \]

2. **Differentiate \( f(x) \)**:
We will differentiate \( f(x) \) term by term. The derivative \( f'(x) \) is given by:
\[ f'(x) = \frac{d}{dx}(x \cos x) + \frac{d}{dx}(\tan x) \]

3. **Differentiate \( \tan x \)**:
The derivative of \( \tan x \) is:
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]

4. **Use the product rule for \( x \cos x \)**:
For the term \( x \cos x \), we apply the product rule, which states that if \( u = x \) and \( v = \cos x \), then:
\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]
Here, \( u = x \) and \( v = \cos x \):
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = -\sin x \)

Applying the product rule:
\[ \frac{d}{dx}(x \cos x) = x(-\sin x) + \cos x(1) = -x \sin x + \cos x \]

5. **Combine the results**:
Now, we can combine the derivatives:
\[ f'(x) = (-x \sin x + \cos x) + \sec^2 x \]

6. **Final expression for the derivative**:
Thus, the derivative of the function \( f(x) \) is:
\[ f'(x) = -x \sin x + \cos x + \sec^2 x \]

### Final Answer:
\[ f'(x) = -x \sin x + \cos x + \sec^2 x \]