`(d)/(dx)(e^((1)/(2)log(1+tan^(2)x)))` is equal to

Options

A. `1/2sec^2x`
B. `sec^2x`
C. `secx.tanx`
D. `e^(1/2log(1+tan^2x)`

Answer

To solve the problem \(\frac{d}{dx}\left(e^{\frac{1}{2}\log(1+\tan^2 x)}\right)\), we will follow these steps:

### Step 1: Simplify the expression inside the derivative
We start with the expression:
\[ e^{\frac{1}{2}\log(1+\tan^2 x)} \]
Using the property of logarithms, we can rewrite this as:
\[ e^{\log((1+\tan^2 x)^{\frac{1}{2}})} \]
This simplifies to:
\[ (1+\tan^2 x)^{\frac{1}{2}} \]

### Step 2: Recognize the identity
We know from trigonometric identities that:
\[ 1 + \tan^2 x = \sec^2 x \]
Thus, we can rewrite our expression as:
\[ \sqrt{1+\tan^2 x} = \sqrt{\sec^2 x} = \sec x \]

### Step 3: Differentiate the simplified expression
Now we need to differentiate \(\sec x\):
\[ \frac{d}{dx}(\sec x) \]
The derivative of \(\sec x\) is given by:
\[ \sec x \tan x \]

### Final Answer
Putting it all together, we have:
\[ \frac{d}{dx}\left(e^{\frac{1}{2}\log(1+\tan^2 x)}\right) = \sec x \tan x \]