`(d)/(dx)tan^(-1) """"(secx + tan x)`=

Options

A. 1
B. `1//2`
C. 2
D. None of these

Answer

To differentiate the function \( y = \tan^{-1}(\sec x + \tan x) \), we will apply the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
The outer function is \( \tan^{-1}(u) \) where \( u = \sec x + \tan x \).

### Step 2: Differentiate the outer function
Using the derivative of \( \tan^{-1}(u) \):
\[ \frac{d}{dx}(\tan^{-1}(u)) = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \]
So, we need to find \( \frac{du}{dx} \).

### Step 3: Differentiate the inner function \( u = \sec x + \tan x \)
Now we differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(\sec x) + \frac{d}{dx}(\tan x) \]
From trigonometric derivatives, we know:
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \quad \text{and} \quad \frac{d}{dx}(\tan x) = \sec^2 x \]
Thus,
\[ \frac{du}{dx} = \sec x \tan x + \sec^2 x \]

### Step 4: Substitute back into the derivative of the outer function
Now we substitute \( u \) and \( \frac{du}{dx} \) back into the derivative:
\[ \frac{dy}{dx} = \frac{1}{1 + (\sec x + \tan x)^2} \cdot (\sec x \tan x + \sec^2 x) \]

### Step 5: Simplify the denominator
We need to simplify \( 1 + (\sec x + \tan x)^2 \):
\[ (\sec x + \tan x)^2 = \sec^2 x + 2\sec x \tan x + \tan^2 x \]
Using the identity \( 1 + \tan^2 x = \sec^2 x \):
\[ 1 + (\sec x + \tan x)^2 = 1 + \sec^2 x + 2\sec x \tan x + \tan^2 x = 1 + \sec^2 x + \tan^2 x + 2\sec x \tan x = 1 + 1 + 2\sec x \tan x = 2 + 2\sec x \tan x \]

### Step 6: Final expression for the derivative
Now substituting back into the derivative:
\[ \frac{dy}{dx} = \frac{\sec x \tan x + \sec^2 x}{2 + 2\sec x \tan x} \]
Factoring out the 2 in the denominator:
\[ \frac{dy}{dx} = \frac{\sec x \tan x + \sec^2 x}{2(1 + \sec x \tan x)} \]

### Step 7: Final simplification
We can factor out \( \sec x \) from the numerator:
\[ \frac{dy}{dx} = \frac{\sec x(\tan x + \sec x)}{2(1 + \sec x \tan x)} \]

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