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
- 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 $\ln(u)$.
- 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 $g(x) = 2x$.
- Differentiate the outer function
Differentiate $f(u) = \ln(u)$. The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$.
$$\frac{d}{du} [\ln(u)] = \frac{1}{u}$$
- Differentiate the inner function
Differentiate $g(x) = 2x$. The derivative of $2x$ with respect to $x$ is 2.
$$\frac{d}{dx} [2x] = 2$$
- 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.