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AP Calculus AB: 4.1.3 Uses of the Power Rule

Mathematics24 CardsCreated 8 months ago

This content highlights the applications of the power rule in calculus, showing how it simplifies the process of finding derivatives. It explains how the rule works with constants (constant multiple rule) and sums of functions (sum rule), and includes examples that combine these rules to handle more complex functions efficiently.

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uses of power rule

• The power rule states that if N is a rational number, then the function
is differentiable and
• Given a differentiable function f and a constant c, the constant multiple rule states that
• Given two differentiable functions f and g, the sum rule states that

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Key Terms

Term
Definition

uses of power rule

• The power rule states that if N is a rational number, then the function
is differentiable and
• Given a differentiable function f and a con...

note

  • The power rule allows you to find the derivative of certain functions without having to use the definition of the derivative.

  • <...

Find the derivative.f(x)=x^4

f’(x)=4x^3

Find the derivative.P(t)=3πt^2

P′(t) = 6 π t

Suppose f(x)=x+2√x+3 3√x.Find f′(x).

f′(x)=1+x^−1/2+x^−2/3

Suppose f(x)=x^2−3x−4. What is the domain of f′(x)?

R

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AP Calculus AB: 4.1.3 Uses of the Power Rule
TermDefinition

uses of power rule

• The power rule states that if N is a rational number, then the function
is differentiable and
• Given a differentiable function f and a constant c, the constant multiple rule states that
• Given two differentiable functions f and g, the sum rule states that

note

  • The power rule allows you to find the derivative of certain functions without having to use the definition of the derivative.

  • To use the power rule, copy the exponent in front of the function and reduce the power by one.

  • Notice that the power rule also works for strange powers such as 1 and 0.

  • Remember, the derivative of a constant function is zero. The derivative of a linear function is a constant.

  • Combining the power rule with other derivative rules makes it even more powerful. One such derivative rule is the constant multiple rule.

  • The constant multiple rule states that the derivative of a
    constant multiplied by a function is equal to the constant
    multiplied by the derivative of the function.

  • The sum rule lets you take the derivative of a function term by term.

  • Notice that you can use the constant multiple rule, the sum rule, and the power rule all together to find a single derivative.

Find the derivative.f(x)=x^4

f’(x)=4x^3

Find the derivative.P(t)=3πt^2

P′(t) = 6 π t

Suppose f(x)=x+2√x+3 3√x.Find f′(x).

f′(x)=1+x^−1/2+x^−2/3

Suppose f(x)=x^2−3x−4. What is the domain of f′(x)?

R

Given that the derivative of √x is(√x)′=1/2√x, find the derivative off(x)=√x/5.

f′(x)=1/10√x.

Find the derivative.f(x)=x^25

25x^24

Suppose a particle’s position is given by f (t) = t ^6 − t ^5 + 1 where t is given in seconds and f (t) is measured in centimeters. What is the velocity of the particle when t = 2?

112 cm/sec

Given that the derivative of 1/x is −1/x^2, find the derivative of f(x)=3/x.

f′(x)=−3/x^2

Given that the derivative of √xis(√x)′=1/2√x, find the derivative off(x)=2√x.

f′(x)=1/√x.

Find the derivative.f(x)=x^3

3x^2

Given that the derivative of 1/x equals −1/x^2,find the derivative of f(x)=−√3/x.

f′(x)=√3/x^2

Suppose f(x)=3x^5−5x^3+2x−6.Find f′(x).

f′(x)=15x^4−15x^2+2

Find the derivative:

f(x)=√3π⋅3√x^4

f’(x)=4/3√3π⋅3√x

Find the derivative.f(x)=3x^8

24x^7

Find the derivative.

p(q)=−π/3√q^3

p′(q)=−π/2 √q

Given that the derivative of 1/x is −1/x^2, find the derivative of f(x)=−5/x.

f′(x)=5/x^2

Find the derivative. f(x)=x^3.15

3.15x^2.15

Suppose f (x) = x^ 6 − x^ 4. Find the equation of the line tangent to f (x) at (1, 0).

y = 2x − 2

Find the derivative. f(x)=2πx^2

f′(x)=4πx

Find the derivative.

f (x) = 2x ^1.45

f ′(x) = 2.9x^ 0.45

Suppose f(x)=x+2√x+3 3√x.Find f′(64).

f′(64)=1 3/16

Suppose f(x)=2x^6+3x^4/3−2/x. Find f′(x).

f′(x)=12x^5+4x^1/3+2x^−2