What Is the Difference Between Derivative of Ln X and Ln |X|?

TL;DR

The derivative of Ln X is 1/X, applicable only for X greater than 0. In contrast, the derivative of Ln |X| is also 1/X but is defined for all X except zero. The key difference lies in their domains, as Ln X is restricted to positive values, while Ln |X| allows for negative values but is undefined at zero.

Transcript

okay media I'm going to talk about the difference between taking the derivative of Ln X versus taking the derivative of Ln of absolute value of x there is the technical detail so let's pay attention to that and to do so they mean just as you get a question right here first suppose I am going to define f of X to be X and let me ask you guys this bas... Read More

Key Insights

• 🥡 Taking the derivative of Ln X and Ln of the absolute value of X both result in 1/X, but with different domain restrictions.
• ☺️ Ln X has a domain of X greater than 0, while Ln of the absolute value of X excludes 0.
• 🥺 The absolute value function is not differentiable at 0, leading to a smaller domain for its derivative.

Questions & Answers

Q: What is the difference between taking the derivative of Ln X and Ln of the absolute value of X?

The main difference is the domain restriction. The derivative of Ln X is 1/X, but X must be greater than 0. In contrast, the derivative of Ln of the absolute value of X is also 1/X, but the domain excludes 0.

Q: Why does the domain of the derivative of Ln of the absolute value of X exclude 0?

The domain excludes 0 because the absolute value of X is not differentiable at 0. This means that the function does not have a tangent line at that point, resulting in a smaller domain for its derivative.

Q: Can Ln of a negative number have a real value?

No, Ln of a negative number does not have any real value. While it has a complex value, the video focuses solely on real numbers.

Q: How is the derivative of the absolute value of X calculated?

The derivative of the absolute value of X is calculated using the chain rule. It is equal to the absolute value of X divided by X. However, the domain of the derivative is smaller than the domain of the original function, excluding 0.

Summary & Key Takeaways

• The video explains that the derivative of Ln X is 1/X, but X must be greater than 0 due to the domain restriction.

• The graph of Ln X is a straight line with a domain of X greater than 0.

• The video also discusses the derivative of Ln of the absolute value of X, which is also 1/X, but with a domain that excludes 0.