What is the derivative of ln(3x)?
Understand the Problem
The question is asking for the derivative of the natural logarithm function applied to 3 times the variable x. We will use the chain rule to find the derivative.
Answer
\frac{1}{x}
Answer for screen readers
The final answer is \frac{1}{x}
Steps to Solve
- Express the function in a more convenient form
The given function is $ ext{ln}(3x) $. First, express it in a simpler way: $ f(x) = ext{ln}(3x) $.
- Apply the properties of logarithms
Use the property of logarithms that $ ext{ln}(a imes b) = ext{ln}(a) + ext{ln}(b) $. In this case: $$ f(x) = ext{ln}(3) + ext{ln}(x) $$
- Differentiate the function
Differentiate $ f(x) $ with respect to $ x $. Recall that $ ext{ln}(3) $ is a constant and the derivative of $ ext{ln}(x) $ with respect to $ x $ is $ rac{1}{x} $: $$ \frac{d}{dx}[ ext{ln}(3) + ext{ln}(x)] = 0 + \frac{1}{x} $$
- Simplify the result
Combine the results from the previous step to get the final answer: $$ \frac{d}{dx} [ ext{ln}(3x)] = \frac{1}{x} $$
The final answer is \frac{1}{x}
More Information
The derivative of a natural logarithm function like ln(3x) shows how the rate of change of the logarithm function behaves with respect to the variable x.
Tips
A common mistake is forgetting the properties of logarithms, particularly that ln(ab) = ln(a) + ln(b).
