derivative of ln(1+1/x), two ways

TL;DR

Learn how to differentiate the function Ln of (1 + 1/X) using two different approaches.

Transcript

okay I'm gonna show you guys how to differentiate the function Ln of 1 plus 1 over X and I'll do it with two ways okay first it's just that I would look at this as how D is and differentiate so let's go ahead and get going I will put on Y prime for that eruptive notation and now we know the derivative of Ln of something it's going to be 1 over the ... Read More

Key Insights

• 📏 Differentiating Ln of (1 + 1/X) can be done using two approaches: one involving the derivative rule and the chain rule, and another involving simplification.
• ❓ Both approaches yield the same result: (-1/X^2) + X.
• 🍽️ The first approach requires the chain rule to differentiate the inner functions, while the second approach simplifies the function before differentiation.
• 💁 Factorizing and rearranging the terms can be done to obtain the same result in a different form.
• 🪡 The derivative of Ln of (1 + X) is simply 1/(1 + X), without needing the chain rule.
• 🪡 The second approach avoids the need for the chain rule by simplifying the function using a common denominator.
• 💨 The two approaches show different ways to arrive at the same derivative result.

Questions & Answers

Q: What is the first approach to differentiating Ln of (1 + 1/X)?

The first approach involves using the derivative rule for Ln and the chain rule to find the derivative. After differentiating, the result is (-1/X^2) + X.

Q: What is the second approach to differentiating Ln of (1 + 1/X)?

The second approach involves simplifying the function by getting a common denominator and then differentiating. The result is also (-1/X^2) + X.

Q: When is the chain rule used in the first approach?

The chain rule is used when differentiating the inside function, which is 1 + 1/X. The derivative of 1 is 0, and the derivative of 1/X is -1/X^2.

Q: Why is the chain rule not needed in the second approach?

In the second approach, the function is simplified before differentiation, removing the need for the chain rule. The derivative of 1 + X is 1, and the derivative of Ln X is 1/X.

Summary & Key Takeaways

• The video teaches how to differentiate the function Ln of (1 + 1/X) using the derivative rule and the chain rule.

• The first approach involves using the derivative rule and the chain rule to find the derivative.

• The second approach involves simplifying the function and applying the derivative rule without needing the chain rule.