# What is the derivative of ln(3x)?

#### Understand the Problem

The question is asking for the derivative of the natural logarithm function ln(3x). To solve this, we can use the chain rule. The derivative of ln(u) is 1/u multiplied by the derivative of u, where u is the function inside the logarithm.

\frac{1}{x}

#### Steps to Solve

1. Identify the function inside the logarithm

The expression inside the natural logarithm is $3x$, so we set $u = 3x$.

1. Differentiate the natural logarithm

Using the chain rule, the derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$. Here, $u = 3x$.

[ \frac{d}{dx}[\ln(3x)] = \frac{1}{3x} \cdot \frac{d}{dx}[3x] ]

1. Calculate the derivative of $3x$

The derivative of $3x$ with respect to $x$ is simply $3$.

[ \frac{d}{dx}[3x] = 3 ]

1. Combine the results

Plug the derivative of $3x$ back into the chain rule formula:

[ \frac{1}{3x} \cdot 3 = \frac{3}{3x} = \frac{1}{x} ]

The derivative of $\ln(3x)$ is $\frac{1}{x}$.

The derivative of $\ln(3x)$ is actually the same as the derivative of $\ln(x)$ because the constant inside the logarithm just introduces a vertical shift in the function, which doesn't affect the slope.