Let `f(x) = int_(0)^(x)(dt)/(sqrt(1+t^(2)))` and `g(x)` be the inverse of `f(x)`, then which one of the following holds good ? (A)` 2g'' = g^(2) ` (B) `2g'' = g^(2)` (C) `3g'' = 2g^(2)` (D) `3g'' = g^(2)`

Options

A. `2g" = g^(2)`
B. `2g" = 3g^(2)`
C. `3g" = 2g^(2)`
D. `3g" = g^(2)`

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined as:

\[ f(x) = \int_0^x \frac{dt}{\sqrt{1+t^2}} \]

and find the properties of its inverse function \( g(x) \), which is defined as \( g(x) = f^{-1}(x) \).

### Step 1: Differentiate \( f(x) \)

Using the Fundamental Theorem of Calculus, we can differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} \left( \int_0^x \frac{dt}{\sqrt{1+t^2}} \right) = \frac{1}{\sqrt{1+x^2}} \]

**Hint:** Remember the Fundamental Theorem of Calculus states that if \( F(x) = \int_a^x f(t) dt \), then \( F'(x) = f(x) \).

### Step 2: Relate \( g(x) \) and \( f(x) \)

Since \( g(x) \) is the inverse of \( f(x) \), we have:

\[ f(g(x)) = x \]

### Step 3: Differentiate the inverse function

Differentiating both sides with respect to \( x \):

\[ f'(g(x)) \cdot g'(x) = 1 \]

From this, we can express \( g'(x) \):

\[ g'(x) = \frac{1}{f'(g(x))} \]

Substituting \( f'(g(x)) \):

\[ g'(x) = \sqrt{1 + g(x)^2} \]

**Hint:** When differentiating an implicit function, use the chain rule and remember that the derivative of \( f(g(x)) \) involves \( f'(g(x)) \cdot g'(x) \).

### Step 4: Differentiate \( g'(x) \) to find \( g''(x) \)

Now, we differentiate \( g'(x) \):

\[ g''(x) = \frac{d}{dx} \left( \sqrt{1 + g(x)^2} \right) \]

Using the chain rule:

\[ g''(x) = \frac{1}{2\sqrt{1 + g(x)^2}} \cdot 2g(x) \cdot g'(x) = \frac{g(x) g'(x)}{\sqrt{1 + g(x)^2}} \]

**Hint:** When differentiating a function that involves a square root, apply the chain rule carefully.

### Step 5: Substitute \( g'(x) \) into \( g''(x) \)

Now substitute \( g'(x) \) back into the equation for \( g''(x) \):

\[ g''(x) = \frac{g(x) \sqrt{1 + g(x)^2}}{\sqrt{1 + g(x)^2}} = g(x) \]

### Conclusion

Thus, we find that:

\[ g''(x) = g(x) \]

This indicates that the second derivative of \( g(x) \) is equal to \( g(x) \) itself.

### Step 6: Match with the options

Now, we check the options provided in the question:

(A) \( 2g'' = g^2 \)
(B) \( 2g'' = g^2 \)
(C) \( 3g'' = 2g^2 \)
(D) \( 3g'' = g^2 \)

None of the options directly match \( g''(x) = g(x) \). However, we can conclude that the correct relation we derived is \( g''(x) = g(x) \).

### Final Answer

Since none of the provided options match our derived equation, we conclude that the answer is not among the options given.