Let `f(x) = tan^(-1)(((x-2))/(x^(2)+2x+2))`,then `26 f'(1)` is

Answer

To find \( 26 f'(1) \) for the function \( f(x) = \tan^{-1}\left(\frac{x-2}{x^2 + 2x + 2}\right) \), we will first need to differentiate \( f(x) \) and then evaluate the derivative at \( x = 1 \).

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

Using the chain rule, the derivative of \( f(x) \) can be expressed as:

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

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

The derivative of \( \tan^{-1}(u) \) is:

\[ \frac{1}{1 + u^2} \cdot \frac{du}{dx} \]

### Step 2: Find \( u^2 \)

First, we need to compute \( u^2 \):

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

Thus,

\[ u^2 = \left(\frac{x-2}{x^2 + 2x + 2}\right)^2 = \frac{(x-2)^2}{(x^2 + 2x + 2)^2} \]

### Step 3: Differentiate \( u \)

Now, we need to find \( \frac{du}{dx} \). Using the quotient rule:

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

Simplifying the numerator:

\[ = (x^2 + 2x + 2) - (2x^2 + 2x - 4) = -x^2 + 6 \]

Thus,

\[ \frac{du}{dx} = \frac{-x^2 + 6}{(x^2 + 2x + 2)^2} \]

### Step 4: Substitute into the derivative formula

Now substituting \( u \) and \( \frac{du}{dx} \) into the derivative of \( f(x) \):

\[ f'(x) = \frac{1}{1 + \frac{(x-2)^2}{(x^2 + 2x + 2)^2}} \cdot \frac{-x^2 + 6}{(x^2 + 2x + 2)^2} \]

This simplifies to:

\[ f'(x) = \frac{(x^2 + 2x + 2)^2}{(x^2 + 2x + 2)^2 + (x-2)^2} \cdot \frac{-x^2 + 6}{(x^2 + 2x + 2)^2} \]

### Step 5: Evaluate \( f'(1) \)

Now, we substitute \( x = 1 \):

1. Calculate \( u \) at \( x = 1 \):

\[ u = \frac{1-2}{1^2 + 2(1) + 2} = \frac{-1}{5} \implies u^2 = \frac{1}{25} \]

2. Calculate \( f'(1) \):

\[ f'(1) = \frac{(1^2 + 2(1) + 2)^2}{(1^2 + 2(1) + 2)^2 + (1-2)^2} \cdot \frac{-1^2 + 6}{(1^2 + 2(1) + 2)^2} \]

Calculating the terms:

\[ 1^2 + 2(1) + 2 = 5 \implies 5^2 = 25 \]

Thus,

\[ f'(1) = \frac{25}{25 + 1} \cdot \frac{5}{25} = \frac{25}{26} \cdot \frac{5}{25} = \frac{5}{26} \]

### Step 6: Calculate \( 26 f'(1) \)

Finally, we find:

\[ 26 f'(1) = 26 \cdot \frac{5}{26} = 5 \]

### Final Answer

\[ \boxed{5} \]