pi-th derivative of x^pi | Summary and Q&A

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December 10, 2019
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blackpenredpen
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pi-th derivative of x^pi

TL;DR

The video explores the derivative of X to the power of PI and introduces a formula for finding the derivative of X to the power of any number.

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Key Insights

  • ✊ The derivative of X to the power of PI is denoted as DX^PI and can be represented as PI! (PI factorial).
  • 🚱 Factorial notation can be extended to non-integer values using the gamma function.
  • #️⃣ The formula for finding the derivative of X to the power of any number involves applying the gamma function to the exponent, subtracting the factorial of the exponent minus the number of times differentiated.
  • 👻 The gamma function is an extension of the factorial function and allows for the calculation of factorial for any real number.
  • ❎ There are some restrictions on the values that can be used in the formula, such as negative numbers.

Transcript

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Questions & Answers

Q: What is the derivative of X to the power of PI?

The derivative of X to the power of PI is denoted as DX^PI and can be represented as PI! (PI factorial).

Q: Can factorial notation be extended to non-integer values?

Yes, factorial notation can be extended to non-integer values using the gamma function, which is represented as gamma(X). It allows for the calculation of factorial for any real number.

Q: How can the gamma function be used to find the derivative of X to the power of any number?

The gamma function can be used to derive a formula for finding the derivative of X to the power of any number. The formula involves applying the gamma function to the exponent and subtracting the factorial of the exponent minus the number of times differentiated.

Q: Are there any restrictions on the values that can be used in the formula?

While the formula works for most real numbers, there are some restrictions, such as negative numbers. Negative numbers do not have a factorial representation and cannot be used in the formula.

Summary & Key Takeaways

  • The video discusses the derivative of X to the power of PI and its connection to factorial notation.

  • It introduces the concept of extending the factorial to non-integer values using the gamma function.

  • A formula for finding the derivative of X to the power of any number is derived using the gamma function.

  • The video encourages viewers to check out the previous videos by Mr. Dutton for more information on fractional derivatives and calculus.

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