4.1.3 Uses of the Power Rule Flashcards Preview

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Flashcards in 4.1.3 Uses of the Power Rule Deck (24)
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1
Q

uses of power rule

A

• 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

2
Q

note

A
  • 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.
3
Q

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

A

f’(x)=4x^3

4
Q

Find the derivative.P(t)=3πt^2

A

P′(t) = 6 π t

5
Q

Suppose f(x)=x+2√x+3 3√x.Find f′(x).

A

f′(x)=1+x^−1/2+x^−2/3

6
Q

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

A

R

7
Q

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

A

f′(x)=1/10√x.

8
Q

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

A

25x^24

9
Q

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?

A

112 cm/sec

10
Q

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

A

f′(x)=−3/x^2

11
Q

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

A

f′(x)=1/√x.

12
Q

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

A

3x^2

13
Q

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

A

f′(x)=√3/x^2

14
Q

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

A

f′(x)=15x^4−15x^2+2

15
Q

Find the derivative:

f(x)=√3π⋅3√x^4

A

f’(x)=4/3√3π⋅3√x

16
Q

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

A

24x^7

17
Q

Find the derivative.

p(q)=−π/3√q^3

A

p′(q)=−π/2 √q

18
Q

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

A

f′(x)=5/x^2

19
Q

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

A

3.15x^2.15

20
Q

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

A

y = 2x − 2

21
Q

Find the derivative.f(x)=2πx^2

A

f′(x)=4πx

22
Q

Find the derivative.

f (x) = 2x ^1.45

A

f ′(x) = 2.9x^ 0.45

23
Q

Suppose f(x)=x+2√x+3 3√x.Find f′(64).

A

f′(64)=1 3/16

24
Q

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

A

f′(x)=12x^5+4x^1/3+2x^−2

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