d/dx ln(2x)
Understand the Problem
The question is asking for the derivative of the function ln(2x) with respect to x. To solve this, we can apply the chain rule and the property of logarithmic differentiation.
Answer
The derivative is $\frac{1}{x}$.
Answer for screen readers
The derivative of the function $\ln(2x)$ with respect to $x$ is $\frac{1}{x}$.
Steps to Solve
- Identify the function for differentiation
We are given the function $f(x) = \ln(2x)$.
- Apply the chain rule
To find the derivative of $f(x)$, we will use the chain rule which states that if $y = \ln(u)$, then the derivative is given by:
$$ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} $$
In this case, $u = 2x$.
- Differentiate the inner function
Now, we need to find the derivative of $u = 2x$. This gives us:
$$ \frac{du}{dx} = 2 $$
- Combine the derivatives
Now we plug the values back into the chain rule formula:
$$ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{2x} \cdot 2 $$
- Simplify the expression
Finally, we simplify:
$$ \frac{dy}{dx} = \frac{2}{2x} = \frac{1}{x} $$
The derivative of the function $\ln(2x)$ with respect to $x$ is $\frac{1}{x}$.
More Information
The derivative represents the rate of change of the natural logarithm function $\ln(2x)$ regarding $x$. In this case, it shows that as $x$ increases, the rate at which $\ln(2x)$ increases follows the simple relation $\frac{1}{x}$. This illustrates how logarithmic functions grow more slowly than linear functions.
Tips
- Not applying the chain rule correctly when differentiating a composite function.
- Forgetting to differentiate the inner function $u = 2x$.
- Incorrectly simplifying the final derivative.
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