If `f(x)=(x+cosx)(x-tanx)," find "f^(')(x)`.

Answer

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

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

### Step-by-Step Solution:

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

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

3. **Differentiate \( v(x) \)**:
\[ v'(x) = \frac{d}{dx}(x - \tan x) = 1 - \sec^2 x \]

4. **Apply the Product Rule**:
Using the product rule:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]

5. **Substitute the Derivatives and Functions**:
\[ f'(x) = (1 - \sin x)(x - \tan x) + (x + \cos x)(1 - \sec^2 x) \]

6. **Simplify the Expression**:
- For the first term:
\[ (1 - \sin x)(x - \tan x) = x - \tan x - x \sin x + \sin x \tan x \]
- For the second term:
\[ (x + \cos x)(1 - \sec^2 x) = (x + \cos x) - (x + \cos x)\sec^2 x \]

7. **Combine and Simplify Further**:
Combine both parts:
\[ f'(x) = (x - \tan x - x \sin x + \sin x \tan x) + (x + \cos x - (x + \cos x)\sec^2 x) \]

8. **Final Expression**:
The final expression for \( f'(x) \) can be simplified further, but this is the general form of the derivative.

### Final Result:
\[ f'(x) = (1 - \sin x)(x - \tan x) + (x + \cos x)(1 - \sec^2 x) \]