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

Options

A. `(5)/(2^(10))`
B. `(1+a^(2))/(a^(10))`
C. `(a^(10))/(1+a^(2))`
D. `(1+a^(10))/(a^(2))`

Answer

To find \( g'(2) \) where \( g \) is the inverse function of \( f \) and given \( f'(x) = \frac{x^{10}}{1+x^2} \) and \( g(2) = a \), we can follow these steps:

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

### Step 2: Differentiate both sides with respect to \( x \)
Using the chain rule, we differentiate:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) = 1 \]

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

### Step 4: Substitute \( x = 2 \)
We need to find \( g'(2) \):
\[ g'(2) = \frac{1}{f'(g(2))} \]
Since \( g(2) = a \), we can substitute:
\[ g'(2) = \frac{1}{f'(a)} \]

### Step 5: Find \( f'(a) \)
We know from the problem that:
\[ f'(x) = \frac{x^{10}}{1+x^2} \]
Thus,
\[ f'(a) = \frac{a^{10}}{1+a^2} \]

### Step 6: Substitute \( f'(a) \) back into the equation for \( g'(2) \)
Now we can write:
\[ g'(2) = \frac{1}{f'(a)} = \frac{1}{\frac{a^{10}}{1+a^2}} = \frac{1+a^2}{a^{10}} \]

### Final Result
Thus, the value of \( g'(2) \) is:
\[ g'(2) = \frac{1 + a^2}{a^{10}} \]