If g(x) is the inverse function of f(x) and `f'(x)=(1)/(1+x^(4))`, then `g'(x)` is

Options

A. `1+[g(x)]^(4)`
B. `1-[g(x)]^(4)`
C. `1+[f(x)]^(4)`
D. `(1)/(1+[g(x)]^(4))`

Explanation

To find \( g'(x) \), where \( g(x) \) is the inverse function of \( f(x) \) and \( f'(x) = \frac{1}{1+x^4} \), we can use the relationship between the derivatives of inverse functions.

### Step-by-Step Solution:

1. **Understand the relationship between \( f \) and \( g \)**:
Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ f(g(x)) = x \]

2. **Differentiate both sides with respect to \( x \)**:
Using the chain rule, we differentiate the left side:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
The right side differentiates to:
\[ \frac{d}{dx}[x] = 1 \]
Therefore, we have:
\[ f'(g(x)) \cdot g'(x) = 1 \]

3. **Solve for \( g'(x) \)**:
Rearranging the equation gives:
\[ g'(x) = \frac{1}{f'(g(x))} \]

4. **Substitute \( f'(x) \)**:
We know that \( f'(x) = \frac{1}{1+x^4} \). Thus, substituting \( g(x) \) into \( f' \):
\[ g'(x) = \frac{1}{f'(g(x))} = \frac{1}{\frac{1}{1+(g(x))^4}} = 1 + (g(x))^4 \]

5. **Final expression for \( g'(x) \)**:
Therefore, we conclude that:
\[ g'(x) = 1 + (g(x))^4 \]

### Final Answer:
\[ g'(x) = 1 + (g(x))^4 \]