Consider a function `f(x)=x^(x), AA x in [1, oo)`. If `g(x)` is the inverse function of `f(x)`, then the value of `g'(4)` is equal to

Options

A. `log_(2)e`
B. `(1)/(2)log_(2e)e`
C. `(1)/(4)log_(2e)e`
D. `(1)/(2)log_(2)2e`

Answer

To solve the problem, we need to find the value of \( g'(4) \) where \( g(x) \) is the inverse of the function \( f(x) = x^x \) for \( x \in [1, \infty) \).

### Step-by-Step Solution:

1. **Understanding the Inverse Function**:
Since \( g(x) \) is the inverse of \( f(x) \), we have the relationship:
\[ g(f(x)) = x \]
Differentiating both sides with respect to \( x \) gives:
\[ g'(f(x)) \cdot f'(x) = 1 \]
Therefore, we can express \( g' \) in terms of \( f' \):
\[ g'(f(x)) = \frac{1}{f'(x)} \]

2. **Finding \( g'(4) \)**:
We need to find \( g'(4) \). This means we need to find \( x \) such that \( f(x) = 4 \). We solve:
\[ x^x = 4 \]
The value of \( x \) that satisfies this equation is \( x = 2 \) since \( 2^2 = 4 \). Thus, \( f(2) = 4 \).

3. **Calculating \( f'(x) \)**:
Next, we need to find \( f'(x) \). The function \( f(x) = x^x \) can be differentiated using logarithmic differentiation:
\[ \ln(f(x)) = x \ln(x) \]
Differentiating both sides:
\[ \frac{f'(x)}{f(x)} = \ln(x) + 1 \]
Therefore,
\[ f'(x) = f(x)(\ln(x) + 1) = x^x(\ln(x) + 1) \]

4. **Finding \( f'(2) \)**:
Now we calculate \( f'(2) \):
\[ f'(2) = 2^2(\ln(2) + 1) = 4(\ln(2) + 1) \]

5. **Calculating \( g'(4) \)**:
Now we substitute \( x = 2 \) into the expression for \( g'(f(x)) \):
\[ g'(4) = g'(f(2)) = \frac{1}{f'(2)} = \frac{1}{4(\ln(2) + 1)} \]

6. **Final Result**:
Thus, the value of \( g'(4) \) is:
\[ g'(4) = \frac{1}{4(\ln(2) + 1)} \]