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 \( f(x) = \tan^{-1}\left(\frac{3x-2}{3+2x}\right) \) for \( x \geq 0 \), we can follow these steps:

### Step 1: Find the derivative of \( f(x) \)
We start with the function:
\[ f(x) = \tan^{-1}\left(\frac{3x-2}{3+2x}\right) \]

To find \( f'(x) \), we use the derivative of the arctangent function:
\[ f'(x) = \frac{1}{1 + \left(\frac{3x-2}{3+2x}\right)^2} \cdot \frac{d}{dx}\left(\frac{3x-2}{3+2x}\right) \]

### Step 2: Differentiate the inner function
Using the quotient rule for differentiation:
\[ \frac{d}{dx}\left(\frac{3x-2}{3+2x}\right) = \frac{(3)(3+2x) - (3x-2)(2)}{(3+2x)^2} \]
Simplifying this gives:
\[ = \frac{9 + 6x - (6x - 4)}{(3+2x)^2} = \frac{9 + 6x - 6x + 4}{(3+2x)^2} = \frac{13}{(3+2x)^2} \]

### Step 3: Substitute back into the derivative of \( f(x) \)
Now substituting back:
\[ f'(x) = \frac{1}{1 + \left(\frac{3x-2}{3+2x}\right)^2} \cdot \frac{13}{(3+2x)^2} \]

### Step 4: Evaluate \( f'(x) \) at \( x = 5 \)
Next, we need to find \( f(5) \):
\[ f(5) = \tan^{-1}\left(\frac{3 \cdot 5 - 2}{3 + 2 \cdot 5}\right) = \tan^{-1}\left(\frac{15 - 2}{3 + 10}\right) = \tan^{-1}\left(\frac{13}{13}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

Now we can find \( f'(5) \):
\[ f'(5) = \frac{1}{1 + \left(\frac{3 \cdot 5 - 2}{3 + 2 \cdot 5}\right)^2} \cdot \frac{13}{(3 + 2 \cdot 5)^2} \]
Calculating \( \left(\frac{13}{13}\right)^2 = 1 \):
\[ f'(5) = \frac{1}{1 + 1} \cdot \frac{13}{(13)^2} = \frac{1}{2} \cdot \frac{13}{169} = \frac{13}{338} \]

### Step 5: Use the relationship between \( g' \) and \( f' \)
Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ g'\left(f(x)\right) = \frac{1}{f'(x)} \]
Thus,
\[ g'\left(\frac{\pi}{4}\right) = \frac{1}{f'(5)} = \frac{1}{\frac{13}{338}} = \frac{338}{13} \]

### Final Answer
Therefore, the value of \( g'\left(\frac{\pi}{4}\right) \) is:
\[ \boxed{26} \]