Find the derivative of x at x=1,

Explanation

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

### Step 1: Identify the function
The function given is:
\[ f(x) = x \]

### Step 2: Use the definition of the derivative
The derivative of a function at a point \( a \) is defined as:
\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]
In this case, we want to find \( f'(1) \).

### Step 3: Substitute into the derivative formula
Substituting \( a = 1 \) into the formula, we get:
\[ f'(1) = \lim_{x \to 1} \frac{f(x) - f(1)}{x - 1} \]

### Step 4: Calculate \( f(1) \)
Now, we need to find \( f(1) \):
\[ f(1) = 1 \]

### Step 5: Substitute \( f(x) \) and \( f(1) \) into the limit
Now, substituting \( f(x) = x \) and \( f(1) = 1 \) into the limit:
\[ f'(1) = \lim_{x \to 1} \frac{x - 1}{x - 1} \]

### Step 6: Simplify the expression
The expression simplifies to:
\[ f'(1) = \lim_{x \to 1} 1 \]
Since the numerator and denominator are the same and cancel out.

### Step 7: Evaluate the limit
Now, we can evaluate the limit:
\[ f'(1) = 1 \]

### Conclusion
Thus, the derivative of \( f(x) = x \) at \( x = 1 \) is:
\[ \boxed{1} \]

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