CramX Logo
RegisterLog in
Back to FlashcardsMathematics / AP Calculus AB: 4.1.1 A Shortcut for Finding Derivatives

AP Calculus AB: 4.1.1 A Shortcut for Finding Derivatives

Mathematics14 CardsCreated 8 months ago

This content introduces the power rule, a shortcut for finding derivatives of functions where variables are raised to rational powers. It explains how to apply the rule and includes examples involving both positive and negative exponents, as well as how to find slopes of tangent lines using derivatives.

Report

Shortcut for Finding Derivatives

  • Using the definition to find the derivative of a function is very time-consuming. However, when dealing with variables raised to rational powers, there is a shortcut you can use that makes finding derivatives easier. This shortcut is called the power rule.

  • The power rule states that if N is a rational number, then the function f(x) = x^N is differentiable and f’(x) = Nx^N-1

Rate to track your progress ✦

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/14

Key Terms

Term
Definition

Shortcut for Finding Derivatives

  • Using the definition to find the derivative of a function is very time-consuming. However, when dealing with variables raised to rational p...

note

  • This is a table of some functions and their derivatives.

  • If you look carefully, you can see a pattern between the
    powers ...

Suppose f(x)=x^−3. What is f′(x)?

f′(x)=−3x^-4

You can use the power rule to take the derivative of functions with exponents expressed as:

  • negative integers

  • natural numbers

  • negative fractions

Suppose f(x)=x^5/2. What is the slope of the line tangent to f at x=4?

20

The power rule is used to find the derivative of what sorts of functions?

Functions of x raised to a power.

Related Flashcard Decks

Mathematics

Apex 2.2.3 Quiz: Graphing Functions

This deck covers key concepts from graphing functions, focusing on understanding graphical representations and their corresponding equations.

9 cards
View Deck
Mathematics

8 Basic Trigonometric Identities

This deck covers the fundamental trigonometric identities, including reciprocal, ratio, and Pythagorean identities.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.1 Antidifferentiation

This content provides an overview of antidifferentiation—the process of finding a function whose derivative is a given function. It introduces the concept of antiderivatives, outlines how to reverse common derivative rules, and includes simple examples to illustrate the idea.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.2 Antiderivatives of Powers

This content explains how to find antiderivatives of functions involving powers of 𝑥, emphasizing the concept of indefinite integrals and the constant of integration 𝐶. It highlights the idea that functions can have infinitely many antiderivatives and includes example problems demonstrating how to evaluate integrals involving powers and rational expressions.

9 cards
View Deck
Mathematics

AP Calculus AB: 9.2.2 Integrating Polynomials by Substitution

This content explains how to integrate polynomial expressions using integration by substitution, especially when dealing with composite functions. It outlines the step-by-step substitution process, emphasizes correct variable handling, and highlights the importance of rewriting the final answer in terms of the original variable along with including the constant of integration.

12 cards
View Deck
Mathematics

AP Calculus AB: 9.3.4 Choosing Effective Function Decompositions

This section focuses on how to strategically choose a substitution when performing integration by substitution. It explains that the best choice for 𝑢 is typically a part of the integrand whose derivative is also present, making the integral easier to evaluate. It also covers how to simplify trigonometric expressions using identities and emphasizes that effective decomposition is key to simplifying complex integrals.

6 cards
View Deck
Mathematics

AP Calculus AB: 9.3.1 Integrating Composite Trigonometric Functions by Substitution

This content covers the technique of integration by substitution applied to composite trigonometric functions. It explains how to identify the inner function for substitution, handle constant multiples, and ensure complete substitution. Worked examples demonstrate how substitution simplifies complex integrals, with emphasis on converting back to the original variable and verifying results through differentiation.

12 cards
View Deck
Mathematics

Conversion Factors and Problem Solving

This deck covers essential concepts related to rounding rules, significant figures, metric conversions, and basic geometry calculations.

18 cards
View Deck
Mathematics

Unit 1 Progress Check: MCQ Part A

This deck covers 18 questions and answers from a unit progress check, focusing on calculus concepts such as limits, derivatives, and function behavior.

18 cards
View Deck
AP Calculus AB: 4.1.1 A Shortcut for Finding Derivatives
TermDefinition

Shortcut for Finding Derivatives

  • Using the definition to find the derivative of a function is very time-consuming. However, when dealing with variables raised to rational powers, there is a shortcut you can use that makes finding derivatives easier. This shortcut is called the power rule.

  • The power rule states that if N is a rational number, then the function f(x) = x^N is differentiable and f’(x) = Nx^N-1

note

  • This is a table of some functions and their derivatives.

  • If you look carefully, you can see a pattern between the
    powers of the terms of the function and the powers of the terms of the derivative.

  • You can also find a pattern between the powers of the terms of the functions and the constants of the terms of the derivatives.

  • In each case, the power of the term of the derivative is one less than the power of the corresponding term of the function.

  • Also, the constant multiple of each term of the derivative is equal to the constant multiple of the corresponding term of the function multiplied by the power of that term.

  • This pattern gives rise to a shortcut called the power rule.

  • The power rule works on any term made up of a variable raised to a rational power.

  • To use the power rule, take the exponent of the original term and multiply it by the term. Then reduce the exponent by one.

Suppose f(x)=x^−3. What is f′(x)?

f′(x)=−3x^-4

You can use the power rule to take the derivative of functions with exponents expressed as:

  • negative integers

  • natural numbers

  • negative fractions

Suppose f(x)=x^5/2. What is the slope of the line tangent to f at x=4?

20

The power rule is used to find the derivative of what sorts of functions?

Functions of x raised to a power.

The power rule can be expressed as:

[xn]′=nx^n−1

When using the power rule, the original coefficient:

Is multiplied by the original exponent

Suppose f(x)=x^7/2.Find the equation of the line tangent to f(x)at (2,8√2).

y=(14√2)x−20√2

Suppose f (x) = x^ −4/3. What is the slope of the line tangent to f at x = 2?

-^3√4 /6

Suppose f (x) = −x ^−1. What is the slope of the line tangent to f at x = 3?

1/9

Suppose f(x)=x7. What is the slope of the line tangent to f at x=2?

448

Find the derivative of f if f (x) = x ^50.

f′(x)=50x^49

When using the power rule, the original exponent:

is reduced by one