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 will use the chain rule of differentiation.
Answer
\( \frac{1}{x} \)
Answer for screen readers
The final answer is ( \frac{1}{x} )
Steps to Solve
- Apply the chain rule
The chain rule states that if you have a composition of functions, you take the derivative of the outer function and multiply it by the derivative of the inner function.
For $f(x) = \ln(3x)$:
$$f(x) = \ln(u) , \text{where} , u = 3x$$
- Differentiate the outer function
Differentiate $\ln(u)$ with respect to $u$:
$$\frac{d}{du}\ln(u) = \frac{1}{u}$$
- Differentiate the inner function
Differentiate $u = 3x$ with respect to $x$:
$$\frac{d}{dx}(3x) = 3$$
- Apply the chain rule to combine the derivatives
Multiply the derivative of the outer function by the derivative of the inner function:
$$\frac{d}{dx}[\ln(3x)] = \frac{1}{3x} \cdot 3 \ = \frac{3}{3x} \ = \frac{1}{x}$$
The final answer is ( \frac{1}{x} )
More Information
The derivative of ( \ln(3x) ) is indeed a common problem where chain rule is nicely applied. The natural logarithm function has fascinating properties, especially when it comes to derivatives, often simplifying results significantly.
Tips
A common mistake is forgetting to apply the chain rule properly, leading to an incorrect multiplier. Always remember to differentiate both the outer and inner functions.
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