# Differentiate ln(2x)

#### Understand the Problem

The question is asking to differentiate the function ln(2x). This involves applying the chain rule and properties of logarithmic differentiation.

$$\frac{1}{x}$$

The final answer is ( \frac{1}{x} )

#### Steps to Solve

1. Identify the components of the function

First, recognize that the function is $\ln(2x)$. This is a composite function where $u = 2x$ and the outer function is $\ln(u)$.

1. Apply the chain rule

The chain rule states that $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$. Here, $f(u) = \ln(u)$ and $g(x) = 2x$.

1. Differentiate the outer function

Differentiate $f(u) = \ln(u)$. The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$.

$$\frac{d}{du} [\ln(u)] = \frac{1}{u}$$

1. Differentiate the inner function

Differentiate $g(x) = 2x$. The derivative of $2x$ with respect to $x$ is 2.

$$\frac{d}{dx} [2x] = 2$$

1. Combine the results using the chain rule

Multiply the derivative of the outer function by the derivative of the inner function.

$$\frac{d}{dx} [\ln(2x)] = \frac{1}{2x} \cdot 2 = \frac{2}{2x} = \frac{1}{x}$$

The final answer is ( \frac{1}{x} )