Power Rule in Differentiation

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson
Download our mobile app to listen on the go
Get App

Questions and Answers

What is the formula for the power rule in differentiation?

f'(x) = nx^(n-1)

What is the derivative of f(x) = x^2 using the power rule?

f'(x) = 2x

How can the power rule be applied to find higher-order derivatives?

The power rule can be applied recursively to find higher-order derivatives.

What is the derivative of f(x) = x using the power rule?

<p>f'(x) = 1</p> Signup and view all the answers

What type of functions can the power rule be used to find the derivative of?

<p>Polynomial functions and functions that involve rational powers</p> Signup and view all the answers

Flashcards are hidden until you start studying

Study Notes

Power Rule

The power rule is a fundamental rule in differentiation that allows us to differentiate functions of the form:

Formula:

If f(x) = x^n, where n is a real number, then f'(x) = nx^(n-1)

Key Points:

  • The power rule can be applied to any function that can be written in the form x^n, where n is a constant.
  • The derivative of x^n is n times x to the power of n-1.
  • The power rule can be applied recursively to find higher-order derivatives.

Examples:

  • If f(x) = x^2, then f'(x) = 2x^(2-1) = 2x
  • If f(x) = x^3, then f'(x) = 3x^(3-1) = 3x^2
  • If f(x) = x^(-2), then f'(x) = (-2)x^(-2-1) = -2x^(-3)

Special Cases:

  • If f(x) = x, then f'(x) = 1x^(1-1) = 1
  • If f(x) = x^0, then f'(x) = 0x^(0-1) = 0 (since any number to the power of 0 is 1, and the derivative of 1 is 0)

Applications:

  • The power rule is used to find the derivative of polynomial functions, which are functions of the form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.
  • The power rule is also used to find the derivative of functions that involve rational powers, such as f(x) = x^(1/2) or f(x) = x^(-3/2).

Power Rule

Formula

  • If f(x) = x^n, where n is a real number, then f'(x) = nx^(n-1).

Key Points

  • The power rule can be applied to any function that can be written in the form x^n, where n is a constant.
  • The derivative of x^n is n times x to the power of n-1.
  • The power rule can be applied recursively to find higher-order derivatives.

Examples

  • If f(x) = x^2, then f'(x) = 2x^(2-1) = 2x.
  • If f(x) = x^3, then f'(x) = 3x^(3-1) = 3x^2.
  • If f(x) = x^(-2), then f'(x) = (-2)x^(-2-1) = -2x^(-3).

Special Cases

  • If f(x) = x, then f'(x) = 1x^(1-1) = 1.
  • If f(x) = x^0, then f'(x) = 0x^(0-1) = 0.

Applications

  • The power rule is used to find the derivative of polynomial functions, which are functions of the form f(x) = a_nx^n + a_(n-1)x^(n-1) +...+ a_1x + a_0.
  • The power rule is also used to find the derivative of functions that involve rational powers, such as f(x) = x^(1/2) or f(x) = x^(-3/2).

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

More Like This

Use Quizgecko on...
Browser
Browser