What is the derivative of `|x-1|` at `x=2` ?

Options

A. `-1`
B. 0
C. 1
D. does not exist

Answer

To find the derivative of the function \( |x-1| \) at \( x=2 \), we can follow these steps:

### Step 1: Understand the absolute value function
The absolute value function \( |x-1| \) can be expressed in piecewise form:
- \( |x-1| = x-1 \) when \( x-1 \geq 0 \) (i.e., \( x \geq 1 \))
- \( |x-1| = -(x-1) = -x + 1 \) when \( x-1 < 0 \) (i.e., \( x < 1 \))

### Step 2: Determine the relevant case for \( x=2 \)
Since \( 2 \geq 1 \), we use the first case:
\[ |x-1| = x-1 \quad \text{for } x \geq 1 \]

### Step 3: Differentiate the function
Now we differentiate \( |x-1| \) in the case where \( x \geq 1 \):
\[ \frac{d}{dx}(x-1) = 1 \]

### Step 4: Evaluate the derivative at \( x=2 \)
Since the derivative of \( |x-1| \) is \( 1 \) for \( x \geq 1 \), we have:
\[ \frac{d}{dx}|x-1| \bigg|_{x=2} = 1 \]

### Final Answer
Thus, the derivative of \( |x-1| \) at \( x=2 \) is:
\[ \boxed{1} \]

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