If `e^f(x)= log x` and g(x) is the inverse function of f(x), then `g'(x)` is

Options

A. `e ^(x) +x`
B. `e ^(e ^(e ^(x)))e ^(e ^(x))`
C. `e ^(e ^(x)+z)`
D. `e ^(e ^(x))`

Answer

To solve the problem, we need to find \( g'(x) \) where \( g(x) \) is the inverse function of \( f(x) \) given that \( e^{f(x)} = \log x \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ e^{f(x)} = \log x \]

2. **Take the natural logarithm of both sides:**
\[ \log(e^{f(x)}) = \log(\log x) \]
Using the property of logarithms, \( \log(e^y) = y \), we simplify:
\[ f(x) = \log(\log x) \]

3. **Since \( g(x) \) is the inverse function of \( f(x) \), we have:**
\[ f(g(x)) = x \]
This means:
\[ \log(\log(g(x))) = x \]

4. **Exponentiate both sides to eliminate the logarithm:**
\[ \log(g(x)) = e^x \]

5. **Exponentiate again to solve for \( g(x) \):**
\[ g(x) = e^{e^x} \]

6. **Now, differentiate \( g(x) \) to find \( g'(x) \):**
\[ g'(x) = \frac{d}{dx}(e^{e^x}) \]
Using the chain rule:
\[ g'(x) = e^{e^x} \cdot \frac{d}{dx}(e^x) \]
The derivative of \( e^x \) is \( e^x \), so:
\[ g'(x) = e^{e^x} \cdot e^x \]

7. **Final result:**
\[ g'(x) = e^{e^x} \cdot e^x \]

### Conclusion:
Thus, the derivative \( g'(x) \) is:
\[ g'(x) = e^{e^x} \cdot e^x \]