What is the derivative of 4 ln x?
Understand the Problem
The question is asking for the derivative of the function 4 ln(x). We will find the derivative using the rules of differentiation.
Answer
The derivative is $\frac{4}{x}$.
Answer for screen readers
The derivative of the function $4 \ln(x)$ is $\frac{4}{x}$.
Steps to Solve
- Identify the function to differentiate
We are given the function to differentiate, which is $f(x) = 4 \ln(x)$.
- Apply the constant multiple rule
The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Therefore, we will take the derivative of $\ln(x)$, which gives us:
$$ f'(x) = 4 \cdot \frac{d}{dx}[\ln(x)] $$
- Differentiate $\ln(x)$
The derivative of $\ln(x)$ is $\frac{1}{x}$. So we substitute this into our earlier expression:
$$ f'(x) = 4 \cdot \frac{1}{x} $$
- Simplify the result
Now we simplify the expression we obtained:
$$ f'(x) = \frac{4}{x} $$
The derivative of the function $4 \ln(x)$ is $\frac{4}{x}$.
More Information
The derivative indicates how the function $4 \ln(x)$ changes with respect to $x$. This result shows that as $x$ increases, the rate of change of the function decreases, which is typical for logarithmic functions.
Tips
- Forgetting to apply the constant multiple rule, which can lead to neglecting the factor of 4.
- Confusing the derivative of $\ln(x)$ with that of other logarithmic bases.
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