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.

Answer

\( \frac{1}{x} \)
Answer for screen readers

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} )

More Information

Differentiating a composite function like ( \ln(2x) ) involves using the chain rule, which is a fundamental concept in calculus.

Tips

A common mistake is to forget to apply the chain rule or to incorrectly differentiate the inner function. Always identify both the inner and outer functions and differentiate them separately before combining the results.

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