Let g (x) be then inverse of f (x) such that `f '(x) =(1)/(1+ x ^(5)),` then ` (d^(2) (g (x)))/(dx ^(2))` is equal to:

Options

A. (a)`(1)/(1+(g(x))^(5))`
B. (b)` (g'(x))/(1+(g(x))^(5))`
C. (c) ` 5 (g (x))^(4) (1+g(x))^(5))`
D. (d)`1+ (g (x))^(5)`

Answer

To solve the problem, we need to find the second derivative of the inverse function \( g(x) \) given that the first derivative of \( f(x) \) is \( f'(x) = \frac{1}{1 + x^5} \).

### Step-by-Step Solution:

1. **Understanding 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**:
Differentiate the equation \( f(g(x)) = x \) with respect to \( x \):
\[ \frac{d}{dx}[f(g(x))] = \frac{d}{dx}[x] \]
By applying the chain rule, we get:
\[ f'(g(x)) \cdot g'(x) = 1 \]

3. **Substituting \( f'(g(x)) \)**:
We know that \( f'(x) = \frac{1}{1 + x^5} \). Therefore, substituting \( g(x) \) into this gives:
\[ f'(g(x)) = \frac{1}{1 + g(x)^5} \]
So, we can rewrite our equation:
\[ \frac{1}{1 + g(x)^5} \cdot g'(x) = 1 \]

4. **Solving for \( g'(x) \)**:
Rearranging the equation to solve for \( g'(x) \):
\[ g'(x) = 1 + g(x)^5 \]

5. **Finding the Second Derivative \( g''(x) \)**:
Differentiate \( g'(x) \) with respect to \( x \):
\[ g''(x) = \frac{d}{dx}[1 + g(x)^5] \]
The derivative of \( 1 \) is \( 0 \), and using the chain rule on \( g(x)^5 \):
\[ g''(x) = 5g(x)^4 \cdot g'(x) \]

6. **Substituting \( g'(x) \) Back**:
Now substitute \( g'(x) \) back into the equation:
\[ g''(x) = 5g(x)^4 \cdot (1 + g(x)^5) \]

### Final Expression:
Thus, the second derivative of \( g(x) \) is:
\[ g''(x) = 5g(x)^4 (1 + g(x)^5) \]