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

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = 1 + \tan x \) and \( v = 1 - \tan x \).

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

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Now, we need to find the derivatives of \( u \) and \( v \):

1. **For \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(1 + \tan x) = 0 + \sec^2 x = \sec^2 x \]

2. **For \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(1 - \tan x) = 0 - \sec^2 x = -\sec^2 x \]

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{(1 - \tan x)(\sec^2 x) - (1 + \tan x)(-\sec^2 x)}{(1 - \tan x)^2} \]

### Step 4: Simplify the Expression
Now, simplify the numerator:

\[ \frac{dy}{dx} = \frac{(1 - \tan x)\sec^2 x + (1 + \tan x)\sec^2 x}{(1 - \tan x)^2} \]

Combine the terms in the numerator:

\[ = \frac{\sec^2 x (1 - \tan x + 1 + \tan x)}{(1 - \tan x)^2} \]

This simplifies to:

\[ = \frac{\sec^2 x (2)}{(1 - \tan x)^2} \]

Thus, we have:

\[ \frac{dy}{dx} = \frac{2 \sec^2 x}{(1 - \tan x)^2} \]

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