(i) ` 4 cot x - 1/2 cos x + 2/cosx - 3/sin x + ( 6 cot x)/(cosecx) + 9`
(ii) ` -5 tan x + 4 tan x cos x -3 cotx sec x + 2 sec x -13`

Answer

To find the derivatives of the given functions, we will apply the rules of differentiation step by step.

### Part (i): Differentiate \( 4 \cot x - \frac{1}{2} \cos x + \frac{2}{\cos x} - \frac{3}{\sin x} + \frac{6 \cot x}{\csc x} + 9 \)

1. **Differentiate each term:**
- The derivative of \( 4 \cot x \) is \( -4 \csc^2 x \).
- The derivative of \( -\frac{1}{2} \cos x \) is \( \frac{1}{2} \sin x \).
- The derivative of \( \frac{2}{\cos x} \) (which is \( 2 \sec x \)) is \( 2 \sec x \tan x \).
- The derivative of \( -\frac{3}{\sin x} \) (which is \( -3 \csc x \)) is \( 3 \csc x \cot x \).
- The derivative of \( \frac{6 \cot x}{\csc x} \) simplifies to \( 6 \cot x \sin x \), and its derivative is \( 6 \left( -\csc^2 x \sin x + \cot x \cos x \right) \).
- The derivative of the constant \( 9 \) is \( 0 \).

2. **Combine the derivatives:**
\[ \frac{d}{dx}(4 \cot x - \frac{1}{2} \cos x + \frac{2}{\cos x} - \frac{3}{\sin x} + \frac{6 \cot x}{\csc x} + 9) = -4 \csc^2 x + \frac{1}{2} \sin x + 2 \sec x \tan x + 3 \csc x \cot x + 6 \left( -\csc^2 x \sin x + \cot x \cos x \right) \]

### Part (ii): Differentiate \( -5 \tan x + 4 \tan x \cos x - 3 \cot x \sec x + 2 \sec x - 13 \)

1. **Differentiate each term:**
- The derivative of \( -5 \tan x \) is \( -5 \sec^2 x \).
- The derivative of \( 4 \tan x \cos x \) requires the product rule:
\[ 4 \left( \sec^2 x \cos x + \tan x (-\sin x) \right) = 4 \sec^2 x \cos x - 4 \tan x \sin x \]
- The derivative of \( -3 \cot x \sec x \) also requires the product rule:
\[ -3 \left( -\csc^2 x \sec x + \cot x \sec x \tan x \right) = 3 \csc^2 x \sec x - 3 \cot x \sec x \tan x \]
- The derivative of \( 2 \sec x \) is \( 2 \sec x \tan x \).
- The derivative of the constant \( -13 \) is \( 0 \).

2. **Combine the derivatives:**
\[ \frac{d}{dx}(-5 \tan x + 4 \tan x \cos x - 3 \cot x \sec x + 2 \sec x - 13) = -5 \sec^2 x + 4 \sec^2 x \cos x - 4 \tan x \sin x + 3 \csc^2 x \sec x - 3 \cot x \sec x \tan x + 2 \sec x \tan x \]

### Final Result
The derivatives for both parts are:
- Part (i):
\[ -4 \csc^2 x + \frac{1}{2} \sin x + 2 \sec x \tan x + 3 \csc x \cot x + 6 \left( -\csc^2 x \sin x + \cot x \cos x \right) \]
- Part (ii):
\[ -5 \sec^2 x + 4 \sec^2 x \cos x - 4 \tan x \sin x + 3 \csc^2 x \sec x - 3 \cot x \sec x \tan x + 2 \sec x \tan x \]