# 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 will use the chain rule of differentiation.

#### Answer

$$\frac{1}{x}$$
##### Answer for screen readers

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

#### Steps to Solve

1. Apply the chain rule

The chain rule states that if you have a composition of functions, you take the derivative of the outer function and multiply it by the derivative of the inner function.

For $f(x) = \ln(3x)$:

$$f(x) = \ln(u) , \text{where} , u = 3x$$

1. Differentiate the outer function

Differentiate $\ln(u)$ with respect to $u$:

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

1. Differentiate the inner function

Differentiate $u = 3x$ with respect to $x$:

$$\frac{d}{dx}(3x) = 3$$

1. Apply the chain rule to combine the derivatives

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

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

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

#### More Information

The derivative of ( \ln(3x) ) is indeed a common problem where chain rule is nicely applied. The natural logarithm function has fascinating properties, especially when it comes to derivatives, often simplifying results significantly.

#### Tips

A common mistake is forgetting to apply the chain rule properly, leading to an incorrect multiplier. Always remember to differentiate both the outer and inner functions.

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