Find the derivative of `y = ln 2x`

Answer

To find the derivative of the function \( y = \ln(2x) \), we can follow these steps:

### Step 1: Rewrite the function using logarithmic properties
Using the property of logarithms that states \( \ln(ab) = \ln(a) + \ln(b) \), we can rewrite the function:
\[ y = \ln(2x) = \ln(2) + \ln(x) \]

### Step 2: Differentiate the function
Now we differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\ln(2)) + \frac{d}{dx}(\ln(x)) \]
Since \( \ln(2) \) is a constant, its derivative is 0. The derivative of \( \ln(x) \) is \( \frac{1}{x} \):
\[ \frac{dy}{dx} = 0 + \frac{1}{x} = \frac{1}{x} \]

### Step 3: Write the final answer
Thus, the derivative of \( y = \ln(2x) \) is:
\[ \frac{dy}{dx} = \frac{1}{x} \]