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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