If `Y = (1 + tan x)/(1 - tan x)`, then `(dy)/(dx)` is

Options

A. `(2 sec^(2)x)/((1- tan x)^(2))`
B. `(2 sec^(2) x)/(1 + tan x)`
C. `("sin"^(2) x)/(1 + tan x)`
D. `(sec^(2) x)/(1 + tan x)`

Answer

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

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

where \( u \) is the numerator and \( v \) is the denominator.

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

### Step 2: Find \( u' \) and \( v' \)
Now, we need to differentiate \( u \) and \( v \):
- The derivative of \( u \):
\[ u' = \frac{d}{dx}(1 + \tan x) = 0 + \sec^2 x = \sec^2 x \]
- The derivative of \( v \):
\[ v' = \frac{d}{dx}(1 - \tan x) = 0 - \sec^2 x = -\sec^2 x \]

### Step 3: Apply the quotient rule
Now, substitute \( u \), \( v \), \( u' \), and \( v' \) into the quotient rule formula:
\[ \frac{dy}{dx} = \frac{(\sec^2 x)(1 - \tan x) - (1 + \tan x)(-\sec^2 x)}{(1 - \tan x)^2} \]

### Step 4: Simplify the expression
Now, simplify the numerator:
\[ \frac{dy}{dx} = \frac{\sec^2 x (1 - \tan x) + \sec^2 x (1 + \tan x)}{(1 - \tan x)^2} \]
Combine the terms in the numerator:
\[ = \frac{\sec^2 x (1 - \tan x + 1 + \tan x)}{(1 - \tan x)^2} = \frac{\sec^2 x (2)}{(1 - \tan x)^2} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{2 \sec^2 x}{(1 - \tan x)^2} \]

### Final Answer
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{2 \sec^2 x}{(1 - \tan x)^2} \]