If y = 2x, find `(dy)/(dx)` from first principles.

Answer

To find the derivative of \( y = 2x \) from first principles, we will use the definition of the derivative. The derivative \( \frac{dy}{dx} \) is defined as:

\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step-by-Step Solution:

1. **Identify the function**:
We have \( y = f(x) = 2x \).

2. **Calculate \( f(x + h) \)**:
Substitute \( x + h \) into the function:
\[ f(x + h) = 2(x + h) = 2x + 2h \]

3. **Set up the difference quotient**:
Now, we need to find \( f(x + h) - f(x) \):
\[ f(x + h) - f(x) = (2x + 2h) - (2x) = 2h \]

4. **Substitute into the limit definition**:
Now substitute this into the limit definition:
\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{2h}{h} \]

5. **Simplify the expression**:
The \( h \) in the numerator and denominator can be cancelled (as long as \( h \neq 0 \)):
\[ \frac{dy}{dx} = \lim_{h \to 0} 2 = 2 \]

6. **Conclusion**:
Therefore, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2 \]