Find the derivative of the following functions :
`2tanx-7secx`

Answer

To find the derivative of the function \( f(x) = 2\tan x - 7\sec x \), we will use the rules of differentiation.

### Step-by-Step Solution:

1. **Identify the function**: We have \( f(x) = 2\tan x - 7\sec x \).

2. **Apply the derivative operator**: We need to find \( \frac{d}{dx}(f(x)) = \frac{d}{dx}(2\tan x) - \frac{d}{dx}(7\sec x) \).

3. **Differentiate each term**:
- For the first term \( 2\tan x \):
- The derivative of \( \tan x \) is \( \sec^2 x \).
- Therefore, \( \frac{d}{dx}(2\tan x) = 2 \cdot \sec^2 x \).
- For the second term \( -7\sec x \):
- The derivative of \( \sec x \) is \( \sec x \tan x \).
- Therefore, \( \frac{d}{dx}(-7\sec x) = -7 \cdot \sec x \tan x \).

4. **Combine the derivatives**:
- Now we combine the results from the derivatives of both terms:
\[ f'(x) = 2\sec^2 x - 7\sec x \tan x \]

5. **Factor out common terms (if possible)**:
- We notice that \( \sec x \) is a common factor in the second term:
\[ f'(x) = 2\sec^2 x - 7\sec x \tan x \]
- We can factor out \( \sec x \) from the expression:
\[ f'(x) = \sec x (2\sec x - 7\tan x) \]

### Final Answer:
Thus, the derivative of the function \( f(x) = 2\tan x - 7\sec x \) is:
\[ f'(x) = \sec x (2\sec x - 7\tan x) \]