If `f(x)=sqrt(x-4sqrt(x-4))+tan^(-1)((1-2x)/(2+x)), AA 4 lt x lt 8`,
then the value of `f'(5)` is equal to

Options

A. `-(7)/(13)`
B. 0
C. `(5)/(13)`
D. `-(8)/(13)`

Answer

To find the value of \( f'(5) \) for the function

\[ f(x) = \sqrt{x - 4\sqrt{x - 4}} + \tan^{-1}\left(\frac{1 - 2x}{2 + x}\right) \]

defined for \( 4 < x < 8 \), we will follow these steps:

### Step 1: Simplify the function \( f(x) \)

We start by rewriting the first term \( \sqrt{x - 4\sqrt{x - 4}} \). We can express it as:

\[ \sqrt{x - 4\sqrt{x - 4}} = \sqrt{(\sqrt{x - 4})^2 - 4(\sqrt{x - 4})} \]

Let \( y = \sqrt{x - 4} \), then \( y^2 = x - 4 \) or \( x = y^2 + 4 \). Substituting this into the equation gives:

\[ \sqrt{y^2 - 4y} = \sqrt{(y - 2)^2} \]

Thus, we have:

\[ \sqrt{x - 4\sqrt{x - 4}} = |y - 2| = 2 - \sqrt{x - 4} \quad \text{(since \( y < 2 \) for \( 4 < x < 8 \))} \]

### Step 2: Rewrite \( f(x) \)

Now we can rewrite \( f(x) \):

\[ f(x) = 2 - \sqrt{x - 4} + \tan^{-1}\left(\frac{1 - 2x}{2 + x}\right) \]

### Step 3: Differentiate \( f(x) \)

Next, we differentiate \( f(x) \):

\[ f'(x) = -\frac{1}{2\sqrt{x - 4}} + \frac{d}{dx} \left(\tan^{-1}\left(\frac{1 - 2x}{2 + x}\right)\right) \]

Using the chain rule for the derivative of \( \tan^{-1}(u) \):

\[ \frac{d}{dx}\left(\tan^{-1}(u)\right) = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \]

where \( u = \frac{1 - 2x}{2 + x} \).

### Step 4: Find \( \frac{du}{dx} \)

Calculating \( \frac{du}{dx} \):

\[ u = \frac{1 - 2x}{2 + x} \]

Using the quotient rule:

\[ \frac{du}{dx} = \frac{(2 + x)(-2) - (1 - 2x)(1)}{(2 + x)^2} \]

Simplifying this gives:

\[ \frac{du}{dx} = \frac{-2(2 + x) - (1 - 2x)}{(2 + x)^2} = \frac{-4 - 2x - 1 + 2x}{(2 + x)^2} = \frac{-5}{(2 + x)^2} \]

### Step 5: Substitute back into \( f'(x) \)

Now substituting back into \( f'(x) \):

\[ f'(x) = -\frac{1}{2\sqrt{x - 4}} - \frac{5}{(2 + x)^2(1 + u^2)} \]

### Step 6: Evaluate \( f'(5) \)

Now we need to evaluate \( f'(5) \):

1. Calculate \( \sqrt{5 - 4} = 1 \).
2. Calculate \( u \) at \( x = 5 \):

\[ u = \frac{1 - 2(5)}{2 + 5} = \frac{1 - 10}{7} = -\frac{9}{7} \]

3. Calculate \( 1 + u^2 \):

\[ 1 + \left(-\frac{9}{7}\right)^2 = 1 + \frac{81}{49} = \frac{49 + 81}{49} = \frac{130}{49} \]

4. Substitute \( x = 5 \) into \( f'(x) \):

\[ f'(5) = -\frac{1}{2 \cdot 1} - \frac{5}{(2 + 5)^2 \cdot \frac{130}{49}} = -\frac{1}{2} - \frac{5}{49 \cdot \frac{130}{49}} = -\frac{1}{2} - \frac{5 \cdot 49}{49 \cdot 130} = -\frac{1}{2} - \frac{5}{130} \]

5. Simplifying gives:

\[ -\frac{1}{2} - \frac{1}{26} = -\frac{13}{26} - \frac{1}{26} = -\frac{14}{26} = -\frac{7}{13} \]

### Final Answer

Thus, the value of \( f'(5) \) is

\[ \boxed{-\frac{7}{13}} \]