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AP Calculus AB: 4.1.1 A Shortcut for Finding Derivatives

Mathematics14 CardsCreated 3 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.

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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

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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.

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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