Let `g` be the inverse function of `f and f'(x)=(x^(10))/(1+x^(2)).` If `f(2)=a` then `g'(2)` is equal to

Options

A. `1/(2^(10))`
B. `(1+a^(2))/(a^(10))`
C. `(a^(10))/(1+a^(2))`
D. `(1+a^(10))/(a^(2))`

Answer

To solve the problem, we need to find \( g'(2) \) where \( g \) is the inverse function of \( f \) and \( f'(x) = \frac{x^{10}}{1+x^2} \). We also know that \( f(2) = a \).

### Step-by-Step Solution:

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

2. **Evaluate \( g(2) \)**:
We know that \( f(2) = a \), which means:
\[ g(2) = f^{-1}(2) = a \]

3. **Substitute \( g(2) \) into the derivative formula**:
We need to find \( g'(2) \):
\[ g'(2) = \frac{1}{f'(g(2))} = \frac{1}{f'(a)} \]

4. **Calculate \( f'(a) \)**:
We have \( f'(x) = \frac{x^{10}}{1+x^2} \). Therefore:
\[ f'(a) = \frac{a^{10}}{1+a^2} \]

5. **Substitute \( f'(a) \) back into the expression for \( g'(2) \)**:
\[ g'(2) = \frac{1}{f'(a)} = \frac{1}{\frac{a^{10}}{1+a^2}} = \frac{1+a^2}{a^{10}} \]

6. **Final expression for \( g'(2) \)**:
Thus, we have:
\[ g'(2) = \frac{1 + a^2}{a^{10}} \]

### Final Answer:
\[ g'(2) = \frac{1 + a^2}{a^{10}} \]