Do you agree? -1/2 ?

TL;DR

The video explains the process of finding the derivatives of Ln x and Ln |x|, highlighting the importance of domain restrictions.

Transcript

that's to summer for fun and this is about derivative in calculus and let me ask you gets the tricky question if the function is Ln X we are trying to find out the derivative of Ln X when X is equal to negative 2 where are you meeting that this is going to be so easy right because of course we can totally differentiate our legs and we differentiate... Read More

Key Insights

• ☺️ Ln X is only defined for X values greater than 0, while Ln |x| has a domain of all real numbers except for X = 0.
• 🙃 The graph of Ln |x| is a mirror image of Ln X, flipping the function to both the right and left sides.
• 🙈 Differentiating Ln |x| requires ignoring the absolute value and treating it as Ln X, with the same derivative of 1/x.
• 🖐️ The domain restrictions play a crucial role in determining the validity and consistency of the function and its derivative.

Questions & Answers

Q: Why is the derivative of Ln X not defined for negative values of X?

The natural logarithm function, Ln X, is only defined for values of X that are greater than 0. Negative values lie outside the domain of the function, resulting in an undefined derivative.

Q: Can X be negative for the function Ln |x|?

Yes, X can be negative for Ln |x|. The absolute value preserves the positive value of negative numbers, allowing them to be included in the domain. However, X cannot be equal to 0, as Ln of 0 is undefined.

Q: How do you find the derivative of Ln |x|?

To find the derivative of Ln |x|, you can ignore the absolute value and differentiate it as you would with Ln x. The derivative for Ln |x| is 1/x, where X is any real number except 0.

Q: What is the significance of the domain restrictions for Ln |x|?

The domain for Ln |x| includes all real numbers except for X = 0. This restriction is necessary because Ln of 0 is undefined. It ensures that the function remains valid and provides consistent results.

Summary & Key Takeaways

• The video discusses finding the derivative of Ln x and emphasizes the need for X to be greater than 0, as Ln X is not defined for negative values.

• It introduces the concept of absolute value and shows the graph of Ln |x|, explaining that X can be negative due to absolute value, but cannot be equal to 0.

• The video demonstrates how to differentiate Ln |x| and clarifies that the domain for this function is all real numbers except for X = 0.