Find the derivative of the functions $y = \ln x$ and $y = \ln 2x$ respectively. Explain why, in (i), having the same derivative does not mean that the two functions are the same.

1. Differentiate $y = \ln x$

For $y = \ln x$,
$\frac{dy}{dx} = \frac{1}{x}$

2. Differentiate $y = \ln 2x$

For $y = \ln 2x$, using the chain rule:
$\frac{dy}{dx} = \frac{1}{2x} \cdot \frac{d}{dx}(2x)$
$\frac{dy}{dx} = \frac{1}{2x} \cdot 2 = \frac{1}{x}$

1. Use laws of logarithms to expand $\ln 2x$

$\ln 2x = \ln 2 + \ln x$

2. Explain the relationship between the two functions and their derivatives

The function $y = \ln 2x$ differs from $y = \ln x$ only by a constant term, $\ln 2$. Since the derivative of any constant is zero, both functions will yield the same derivative, even though they are not the same function.