If `f (x) =|x-5| and g (x) =f (f (x)) AA x gt 10,` then g' (x) equals

Options

A. 1
B. `-1`
C. 0
D. None of these

Answer

To solve the problem step by step, we will analyze the functions given and find the derivative of \( g(x) \).

### Step 1: Define the functions
We have:
- \( f(x) = |x - 5| \)
- \( g(x) = f(f(x)) \)

### Step 2: Analyze \( f(x) \)
Since we are given that \( x > 10 \), we can simplify \( f(x) \):
- For \( x > 10 \), \( x - 5 > 0 \) so \( f(x) = x - 5 \).

### Step 3: Find \( g(x) \)
Now we need to find \( g(x) = f(f(x)) \):
1. First, calculate \( f(x) \):
\[ f(x) = x - 5 \]
2. Now, substitute \( f(x) \) into itself:
\[ g(x) = f(f(x)) = f(x - 5) \]
3. Since \( x - 5 > 5 \) when \( x > 10 \), we have:
\[ f(x - 5) = (x - 5) - 5 = x - 10 \]
Thus, we can write:
\[ g(x) = x - 10 \]

### Step 4: Differentiate \( g(x) \)
Now, we differentiate \( g(x) \):
\[ g'(x) = \frac{d}{dx}(x - 10) \]
The derivative of \( x \) is \( 1 \) and the derivative of a constant \( -10 \) is \( 0 \):
\[ g'(x) = 1 \]

### Conclusion
The value of \( g'(x) \) is \( 1 \).

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