The Derivative of f(x) = |ln(x)|

TL;DR

The video explains how to find the derivative of the absolute value of the natural log of X and highlights the restrictions on the function.

Transcript

hey YouTube in this video we're going to find the derivative of the absolute value of the natural log of X solution so we have f of X and it's equal to the absolute value of the natural log of X and we want to take the derivative obviously this will not be differentiable everywhere so at the end of the problem we'll talk about the restrictions so t... Read More

Key Insights

• 😑 The absolute value of X can be represented as X squared to the one-half power, which simplifies the expression.
• 📏 The chain rule is utilized to find the derivative of the given function.
• 😑 The derivative is expressed as 1 over X times the natural log of X divided by the absolute value of the natural log of X.

Questions & Answers

Q: How can the absolute value of X be written in terms of Ln X?

The absolute value of X can be written as the square root of Ln X quantity squared.

Q: What are the steps to find the derivative of the given function?

The derivative is found by using the chain rule and simplifying the expression. The steps include differentiating the outer function, multiplying by the derivative of the inner function, and canceling out terms, resulting in 1 over X times the natural log of X divided by the absolute value of the natural log of X.

Q: What are the restrictions on the function?

The restrictions are that X cannot be equal to 0 or 1, as the natural log of 0 is undefined and the natural log of 1 is 0. Therefore, Ln X cannot be equal to 0 for all values of X.

Q: Why is the chain rule used in finding the derivative?

The chain rule is used because the function involves nested functions. By applying the chain rule, the derivative of the outer function is found while taking into account the derivative of the inner function.

Summary & Key Takeaways

• The absolute value of X can be written as X squared to the one-half power, which simplifies to the square root of Ln X quantity squared.

• Using the chain rule, the derivative of the function is found to be 1 over X times the natural log of X divided by the absolute value of the natural log of X.

• Restrictions on the function include X not being equal to 0 or 1, as the natural log of 0 is undefined and the natural log of 1 is 0.