Find the derivative of x using first principle method.

Answer

To find the derivative of \( f(x) = x \) using the first principle method, we will follow these steps:

### Step 1: Define the function
Let \( f(x) = x \).

### Step 2: Use the definition of the derivative
The derivative of \( f(x) \) using the first principle is given by:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step 3: Substitute the function into the formula
Substituting \( f(x) = x \) into the formula, we get:
\[ f'(x) = \lim_{h \to 0} \frac{(x + h) - x}{h} \]

### Step 4: Simplify the expression
Now simplify the expression inside the limit:
\[ f'(x) = \lim_{h \to 0} \frac{x + h - x}{h} = \lim_{h \to 0} \frac{h}{h} \]

### Step 5: Cancel the \( h \) terms
Since \( h \) is not equal to zero (as we are taking the limit), we can cancel \( h \):
\[ f'(x) = \lim_{h \to 0} 1 \]

### Step 6: Evaluate the limit
Now, evaluate the limit:
\[ f'(x) = 1 \]

### Conclusion
Thus, the derivative of \( f(x) = x \) is:
\[ f'(x) = 1 \]

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