Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers) :
`(secx-1)/(secx+1)`

Answer

To find the derivative of the function \( y = \frac{\sec x - 1}{\sec x + 1} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{u'v - uv'}{v^2} \]

where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives.

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = \sec x - 1 \)
- \( v = \sec x + 1 \)

### Step 2: Find the derivatives \( u' \) and \( v' \)
We need to find the derivatives of \( u \) and \( v \):
- The derivative of \( u \):
\[ u' = \frac{d}{dx}(\sec x - 1) = \sec x \tan x - 0 = \sec x \tan x \]

- The derivative of \( v \):
\[ v' = \frac{d}{dx}(\sec x + 1) = \sec x \tan x + 0 = \sec x \tan x \]

### Step 3: Apply the quotient rule
Now, we can apply the quotient rule:
\[ \frac{dy}{dx} = \frac{(\sec x \tan x)(\sec x + 1) - (\sec x - 1)(\sec x \tan x)}{(\sec x + 1)^2} \]

### Step 4: Simplify the numerator
Let's simplify the numerator:
1. Expand the terms:
- First term: \( \sec x \tan x (\sec x + 1) = \sec^2 x \tan x + \sec x \tan x \)
- Second term: \( (\sec x - 1)(\sec x \tan x) = \sec^2 x \tan x - \sec x \tan x \)

2. Combine the terms:
\[ \sec^2 x \tan x + \sec x \tan x - (\sec^2 x \tan x - \sec x \tan x) = \sec^2 x \tan x + \sec x \tan x - \sec^2 x \tan x + \sec x \tan x \]
This simplifies to:
\[ 2 \sec x \tan x \]

### Step 5: Write the final derivative
Now, substituting back into the derivative expression:
\[ \frac{dy}{dx} = \frac{2 \sec x \tan x}{(\sec x + 1)^2} \]

### Final Result
Thus, the derivative of the function \( y = \frac{\sec x - 1}{\sec x + 1} \) is:
\[ \frac{dy}{dx} = \frac{2 \sec x \tan x}{(\sec x + 1)^2} \]