What is the derivative of ln n?
Understand the Problem
The question is asking for the mathematical derivative of the natural logarithm of n, expressed as ln(n). To solve this, we will apply the rules of differentiation in calculus.
Answer
The derivative of $\ln(n)$ is $\frac{1}{n}$.
Answer for screen readers
The derivative of $\ln(n)$ with respect to $n$ is $\frac{1}{n}$.
Steps to Solve
- Identify the function to differentiate
We are given the function $f(n) = \ln(n)$ which we want to differentiate with respect to $n$.
- Use the derivative rule for logarithmic functions
The derivative of the natural logarithm function $\ln(n)$ can be expressed using a standard rule in calculus: $$ f'(n) = \frac{1}{n} $$
- State the result of differentiation
Thus, when we differentiate $f(n) = \ln(n)$, we arrive at: $$ \frac{d}{dn}(\ln(n)) = \frac{1}{n} $$
The derivative of $\ln(n)$ with respect to $n$ is $\frac{1}{n}$.
More Information
The derivative of the natural logarithm is a fundamental concept in calculus and is widely used in many fields including science, engineering, and economics. It indicates the rate at which the function $\ln(n)$ changes as $n$ changes.
Tips
- A common mistake is to confuse the derivative of $\ln(n)$ with other logarithmic forms, such as $\log_{10}(n)$, which has a different derivative.
- Another mistake is forgetting that the derivative only applies when $n > 0$, as $\ln(n)$ is only defined for positive values of $n$.
