derivative of sin^-1(2x+1), calculus 1 tutorial

TL;DR

This content explains how to simplify the derivative of inverse sine and demonstrates a step-by-step process.

Transcript

3.5 number 51 we are going to differentiate why it's equal to inverse sine of 2x plus 1 and remember that the derivative of inverse sine X is equal to 1 over square root of 1 minus x squared right because of that we can get work I will use dy/dx as my notation here or the derivative is equal to so the outside function sign is the inverse sine of so... Read More

Key Insights

• ☺️ The derivative of inverse sine x is equal to 1 over the square root of 1 minus x squared.
• 😑 To simplify the derivative of inverse sine, utilize the chain rule and simplify the expression inside the square root.
• 🧑‍🏭 By factoring out common factors and simplifying further, the derivative can be simplified to a simpler form.
• 😑 Understanding the properties of square roots and factoring can help in simplifying mathematical expressions.
• ❓ The step-by-step process demonstrated in the content can be applied to similar differentiation problems.
• ☠️ Differentiation is a fundamental concept in calculus that helps find rates of change and slopes of curves.
• 👻 The chain rule is an important tool in calculus that allows us to differentiate composite functions.

Questions & Answers

Q: How can we differentiate 3.5 number 51 and simplify it?

To differentiate and simplify 3.5 number 51, we can use the chain rule and simplify the expression inside the square root to get the square root of negative x squared minus x.

Q: Why is the derivative of inverse sine x equal to 1 over the square root of 1 minus x squared?

The derivative of inverse sine x is derived from the chain rule and is equal to 1 over the square root of 1 minus x squared. This is because the derivative of the inside function is 2x plus 1, and the derivative of inverse sine x is 1 over the square root of 1 minus x squared.

Q: How do we simplify the expression inside the square root?

To simplify the expression inside the square root, we square (2x + 1) to get 4x squared plus 4x plus 1. Then, we can rewrite it as 1 minus 4x squared minus 4x minus 1. Simplifying further, we get 4 times the square root of negative x squared minus x.

Q: Why do we factor out the 4 inside the square root?

We factor out the 4 inside the square root to simplify the expression further. By factoring out the 4, we can rewrite it as the square root of 4 times the square root of negative x squared minus x. The square root of 4 is 2.

Summary & Key Takeaways

• The content discusses differentiating 3.5 number 51 and shows how it is equal to the inverse sine of 2x plus 1.

• The derivative of inverse sine x is explained and used to simplify the expression further.

• By utilizing the chain rule and simplifying the inside expression, the derivative is simplified to the square root of negative x squared minus x.