If `f'(x)=arc tan((x^(x)-x^(-x))/(2))`, then f'(1) is equal to

Options

A. 1
B. -1
C. `log2`
D. none of these

Answer

To find \( f'(1) \) given that \( f'(x) = \tan^{-1}\left(\frac{x^x - x^{-x}}{2}\right) \), we will follow these steps:

### Step 1: Substitute \( x = 1 \) into the expression for \( f'(x) \)

We start with the expression for \( f'(x) \):
\[ f'(x) = \tan^{-1}\left(\frac{x^x - x^{-x}}{2}\right) \]

Now, substitute \( x = 1 \):
\[ f'(1) = \tan^{-1}\left(\frac{1^1 - 1^{-1}}{2}\right) \]

### Step 2: Evaluate \( 1^1 \) and \( 1^{-1} \)

Calculating the powers:
\[ 1^1 = 1 \]
\[ 1^{-1} = \frac{1}{1} = 1 \]

### Step 3: Substitute these values back into the expression

Now, substitute these values back into the equation:
\[ f'(1) = \tan^{-1}\left(\frac{1 - 1}{2}\right) \]

### Step 4: Simplify the expression

Simplifying the fraction:
\[ f'(1) = \tan^{-1}\left(\frac{0}{2}\right) = \tan^{-1}(0) \]

### Step 5: Evaluate \( \tan^{-1}(0) \)

The value of \( \tan^{-1}(0) \) is:
\[ \tan^{-1}(0) = 0 \]

### Conclusion

Thus, we find that:
\[ f'(1) = 0 \]