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 function of \( f(x) = x^x \) for \( x \in [1, \infty) \).

### Step-by-Step Solution:

1. **Understand the relationship between \( g \) and \( f \)**:
Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ g(f(x)) = x \]
Differentiating both sides with respect to \( x \) gives:
\[ g'(f(x)) \cdot f'(x) = 1 \]
Therefore,
\[ g'(f(x)) = \frac{1}{f'(x)} \]

2. **Calculate \( f'(x) \)**:
We start with \( f(x) = x^x \). To differentiate \( f(x) \), we can use logarithmic differentiation:
\[ y = x^x \implies \log y = x \log x \]
Differentiating both sides:
\[ \frac{1}{y} \frac{dy}{dx} = \log x + 1 \]
Thus,
\[ \frac{dy}{dx} = y(\log x + 1) = x^x (\log x + 1) \]
Therefore, we have:
\[ f'(x) = x^x (\log x + 1) \]

3. **Find \( g'(4) \)**:
We need to find \( g'(4) \). First, we need to determine \( x \) such that \( f(x) = 4 \):
\[ x^x = 4 \]
We can find that \( x = 2 \) satisfies this equation since \( 2^2 = 4 \). Therefore, \( f(2) = 4 \).

4. **Substituting back to find \( g'(4) \)**:
Now, substituting \( x = 2 \) into the expression for \( g' \):
\[ g'(4) = g'(f(2)) = \frac{1}{f'(2)} \]
We need to calculate \( f'(2) \):
\[ f'(2) = 2^2 (\log 2 + 1) = 4(\log 2 + 1) \]
Thus,
\[ g'(4) = \frac{1}{4(\log 2 + 1)} \]

5. **Final expression**:
Therefore, the value of \( g'(4) \) is:
\[ g'(4) = \frac{1}{4(\log 2 + 1)} \]