Consider the function `f(x)=tan^(-1){(3x-2)/(3+2x)}, AA x ge 0.` If `g(x)` is the inverse function of `f(x)`, then the value of `g'((pi)/(4))` is equal to

Answer

To find the value of \( g'\left(\frac{\pi}{4}\right) \) where \( g(x) \) is the inverse function of \( f(x) = \tan^{-1}\left(\frac{3x - 2}{3 + 2x}\right) \), we can follow these steps:

### Step 1: Understand the relationship between \( f \) and \( g \)

Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ f(g(x)) = x \]
Differentiating both sides with respect to \( x \) gives us:
\[ f'(g(x)) \cdot g'(x) = 1 \]
Thus, we can express \( g'(x) \) as:
\[ g'(x) = \frac{1}{f'(g(x))} \]

### Step 2: Find \( g\left(\frac{\pi}{4}\right) \)

We need to find \( g\left(\frac{\pi}{4}\right) \). This means we need to find \( x \) such that:
\[ f(x) = \frac{\pi}{4} \]
This leads us to solve:
\[ \tan^{-1}\left(\frac{3x - 2}{3 + 2x}\right) = \frac{\pi}{4} \]
Taking the tangent of both sides:
\[ \frac{3x - 2}{3 + 2x} = 1 \]

### Step 3: Solve for \( x \)

Cross-multiplying gives:
\[ 3x - 2 = 3 + 2x \]
Rearranging the equation:
\[ 3x - 2x = 3 + 2 \]
\[ x = 5 \]
Thus, \( g\left(\frac{\pi}{4}\right) = 5 \).

### Step 4: Find \( f'(x) \)

Next, we need to find \( f'(x) \). The function is:
\[ f(x) = \tan^{-1}\left(\frac{3x - 2}{3 + 2x}\right) \]
Using the derivative of \( \tan^{-1}(u) \), we have:
\[ f'(x) = \frac{1}{1 + \left(\frac{3x - 2}{3 + 2x}\right)^2} \cdot \frac{d}{dx}\left(\frac{3x - 2}{3 + 2x}\right) \]

### Step 5: Differentiate \( \frac{3x - 2}{3 + 2x} \)

Using the quotient rule:
\[ \frac{d}{dx}\left(\frac{3x - 2}{3 + 2x}\right) = \frac{(3)(3 + 2x) - (3x - 2)(2)}{(3 + 2x)^2} \]
Calculating the numerator:
\[ 9 + 6x - (6x - 4) = 9 + 6x - 6x + 4 = 13 \]
Thus:
\[ \frac{d}{dx}\left(\frac{3x - 2}{3 + 2x}\right) = \frac{13}{(3 + 2x)^2} \]

### Step 6: Substitute back into \( f'(x) \)

Now substituting back:
\[ f'(x) = \frac{1}{1 + \left(\frac{3x - 2}{3 + 2x}\right)^2} \cdot \frac{13}{(3 + 2x)^2} \]

### Step 7: Evaluate \( f'(5) \)

Substituting \( x = 5 \):
\[ f'(5) = \frac{1}{1 + \left(\frac{3(5) - 2}{3 + 2(5)}\right)^2} \cdot \frac{13}{(3 + 2(5))^2} \]
Calculating:
\[ \frac{3(5) - 2}{3 + 2(5)} = \frac{15 - 2}{3 + 10} = \frac{13}{13} = 1 \]
Thus:
\[ f'(5) = \frac{1}{1 + 1^2} \cdot \frac{13}{(3 + 10)^2} = \frac{1}{2} \cdot \frac{13}{169} = \frac{13}{338} \]

### Step 8: Find \( g'\left(\frac{\pi}{4}\right) \)

Finally, substituting into the expression for \( g' \):
\[ g'\left(\frac{\pi}{4}\right) = \frac{1}{f'(5)} = \frac{1}{\frac{13}{338}} = \frac{338}{13} \]

### Final Answer

Thus, the value of \( g'\left(\frac{\pi}{4}\right) \) is:
\[ \boxed{\frac{338}{13}} \]