How to Find the Derivative of x^(2/3) Using Definition

TL;DR

To find the derivative of the function x^(2/3) using the definition, apply the limit of the difference quotient, simplifying by factoring the difference of cubes. Ultimately, the derivative is 2/3 * x^(-1/3), which can also be expressed as 2/(3 * cube root of x).

Transcript

okay mike this is for you i will show you how to use the definition of derivative to find the gravity of this function and of course at the end we will talk about the power rule as well so let's focus on the definition of derivative we know that if this is the function then f prime of x is by definition we will have the limit as h goes to zero and ... Read More

Key Insights

• 😑 The derivative of the cube root function can be found using the definition of the derivative and simplifying algebraic expressions.
• 🧊 Factoring the difference of two cubes is a useful technique in simplifying expressions involving cube roots.
• 😑 The power rule can be used to simplify the expression of the derivative of the cube root function.

Questions & Answers

Q: How is the definition of the derivative used to find the derivative of the cube root function?

The definition of the derivative is used to find the derivative of the cube root function by taking the limit as h approaches zero of the difference quotient. This involves evaluating the function at two points and finding the slope of the secant line between them.

Q: How is the power rule used to simplify the expression?

The power rule is used to simplify the expression of the derivative of the cube root function. By writing the expression as x to the power of 2/3, we can bring the exponent to the front and subtract one from it. This results in a simplified expression.

Q: What is the purpose of factoring the difference of two cubes?

Factoring the difference of two cubes allows us to simplify the expression and cancel out common terms. This simplification helps in finding the derivative of the cube root function more efficiently.

Q: Why is it important to pay attention to the teachers and not rely solely on the power rule?

It is important to pay attention to the teachers and not rely solely on the power rule because calculus involves more than just using the power rule. There are many other concepts and techniques that need to be understood and applied correctly.

Summary & Key Takeaways

• The video explains the process of finding the derivative of the cube root function using the definition of the derivative.

• The video demonstrates how to factor the difference of two cubes to simplify the expression.

• The video shows how to simplify the expression further by canceling out common terms.