What is the derivative of ln(3x)?
Understand the Problem
The question is asking for the derivative of the natural logarithm function ln(3x). To solve this, we can use the chain rule. The derivative of ln(u) is 1/u multiplied by the derivative of u, where u is the function inside the logarithm.
Answer
\frac{1}{x}
Answer for screen readers
The final answer is \frac{1}{x}
Steps to Solve
- Identify the function inside the logarithm
The expression inside the natural logarithm is $3x$, so we set $u = 3x$.
- Differentiate the natural logarithm
Using the chain rule, the derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$. Here, $u = 3x$.
[ \frac{d}{dx}[\ln(3x)] = \frac{1}{3x} \cdot \frac{d}{dx}[3x] ]
- Calculate the derivative of $3x$
The derivative of $3x$ with respect to $x$ is simply $3$.
[ \frac{d}{dx}[3x] = 3 ]
- Combine the results
Plug the derivative of $3x$ back into the chain rule formula:
[ \frac{1}{3x} \cdot 3 = \frac{3}{3x} = \frac{1}{x} ]
The derivative of $\ln(3x)$ is $\frac{1}{x}$.
The final answer is \frac{1}{x}
More Information
The derivative of $\ln(3x)$ is actually the same as the derivative of $\ln(x)$ because the constant inside the logarithm just introduces a vertical shift in the function, which doesn't affect the slope.
Tips
A common mistake is to forget to multiply by the derivative of the inner function. Always remember to apply the chain rule completely.