4.1.3 Uses of the Power Rule

Question 1

uses of power rule

Answer 1

• 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

Question 2

note

Answer 2

• 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.

Question 3

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

Answer 3

f’(x)=4x^3

Question 4

Find the derivative.P(t)=3πt^2

Answer 4

P′(t) = 6 π t

Question 5

Suppose f(x)=x+2√x+3 3√x.Find f′(x).

Answer 5

f′(x)=1+x^−1/2+x^−2/3

Question 6

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

Answer 6

R

Question 7

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

Answer 7

f′(x)=1/10√x.

Question 8

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

Answer 8

25x^24

Question 9

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?

Answer 9

112 cm/sec

Question 10

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

Answer 10

f′(x)=−3/x^2

Question 11

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

Answer 11

f′(x)=1/√x.

Question 12

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

Answer 12

3x^2

Question 13

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

Answer 13

f′(x)=√3/x^2

Question 14

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

Answer 14

f′(x)=15x^4−15x^2+2

Question 15

Find the derivative:

f(x)=√3π⋅3√x^4

Answer 15

f’(x)=4/3√3π⋅3√x

Question 16

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

Answer 16

24x^7

Question 17

Find the derivative.

p(q)=−π/3√q^3

Answer 17

p′(q)=−π/2 √q

Question 18

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

Answer 18

f′(x)=5/x^2

Question 19

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

Answer 19

3.15x^2.15

Question 20

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

Answer 20

y = 2x − 2

Question 21

Find the derivative.f(x)=2πx^2

Answer 21

f′(x)=4πx

Question 22

Find the derivative.

f (x) = 2x ^1.45

Answer 22

f ′(x) = 2.9x^ 0.45

Question 23

Suppose f(x)=x+2√x+3 3√x.Find f′(64).

Answer 23

f′(64)=1 3/16

Question 24

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

Answer 24

f′(x)=12x^5+4x^1/3+2x^−2