Differentiate from first principles:
1. `2x`

Answer

To differentiate the function \( f(x) = 2x \) from first principles, we will follow these steps:

### Step 1: Write the definition of the derivative using first principles.
The derivative of a function \( f(x) \) from first principles is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 2: Substitute \( f(x) \) into the formula.
Given \( f(x) = 2x \), we need to find \( f(x+h) \):
\[ f(x+h) = 2(x+h) = 2x + 2h \]
Now substitute \( f(x+h) \) and \( f(x) \) into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{(2x + 2h) - (2x)}{h} \]

### Step 3: Simplify the expression.
Now simplify the expression inside the limit:
\[ f'(x) = \lim_{h \to 0} \frac{2x + 2h - 2x}{h} = \lim_{h \to 0} \frac{2h}{h} \]

### Step 4: Cancel out \( h \).
Since \( h \neq 0 \) (as we are taking the limit), we can cancel \( h \):
\[ f'(x) = \lim_{h \to 0} 2 = 2 \]

### Step 5: State the result.
Thus, the derivative of the function \( f(x) = 2x \) is:
\[ f'(x) = 2 \]