Derivatives: Definition, Rules, and Applications

Question 1

What is the derivative of the function g(x) = e^(-x)?

Accepted Answer

-e^(-x)

Answer

e^(-x)

Answer

-x*e^(-x)

Answer

x*e^(-x)

Question 2

At what value(s) of x does the function f(x) = x^3 - 6x^2 + 9x have a relative minimum?

Accepted Answer

x = 0 and x = 3

Answer

x = 0

Answer

x = 1

Answer

x = 3

Question 3

Which of the following functions has the least rate of change at x = 1?

Accepted Answer

k(x) = x

Answer

f(x) = x^4

Answer

g(x) = x^3

Answer

h(x) = x^2

Question 4

Calculate the derivative of the function f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² + 2x - 5

Answer

x² + 2x - 5

Question 5

Determine the derivative of the function f(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(4x)

Answer

2sin(2x)

Question 6

Calculate the derivative of the function f(x) = e^x * ln(x).

Accepted Answer

e^x * ln(x) + e^x/(x * ln(x))

Answer

e^x * (1/x)

Answer

e^x * ln(x) - e^x/(x * ln(x))

Answer

e^x * (1 + ln(x))

Question 7

Determine the derivative of the function f(x) = 1 / (x² + 1).

Accepted Answer

-2x / (x² + 1)²

Answer

2x / (x² + 1)²

Answer

-1 / (x² + 1)

Answer

1 / (x² + 1)

Question 8

Find the derivative of the function f(x) = e^(2x).

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2xe^(2x)

Answer

(1/2)e^(2x)

Question 9

Calculate the derivative of f(x) = ln(x^2).

Accepted Answer

(2/x)ln(x)

Answer

(2x)ln(x)

Answer

(1/x)ln(x)

Answer

(2x^2)ln(x)

Question 10

Find the derivative of the function f(x) = x^3 * sin(x).

Accepted Answer

3x^2 * sin(x) + x^3 * cos(x)

Answer

x^2 * sin(x) + x^3 * cos(x)

Answer

x^3 * cos(x) - sin(x)

Answer

x^3 * cos(x) + sin(x)

Question 11

At the point (1, 3), determine the slope of the tangent line to the curve y = x^2 + 2x.

Accepted Answer

5

Answer

1

Answer

3

Answer

7

Question 12

Precisely, what is the definition of the derivative of a function f(x)?

Accepted Answer

The limit of the average rate of change of f(x) as the change in x approaches zero

Answer

The slope of the tangent line to the graph of f(x) at a given point

Answer

The instantaneous rate of change of f(x)

Answer

The area under the curve of f(x)

Question 13

Calculate the derivative of the function f(x) = x^3 + 2x^2 - 5.

Accepted Answer

3x^2 + 4x

Answer

2x^3 + 4x

Answer

3x^3 - 4x

Answer

2x^2 - 5

Question 14

What is the formal definition of the derivative of a function f(x)?

Accepted Answer

The instantaneous rate of change of f(x) with respect to x

Answer

The slope of the tangent line to the graph of f(x) at a given point

Answer

The area under the curve of f(x) over a given interval

Answer

The maximum or minimum value of f(x)

Question 15

Employing the power rule, evaluate the derivative of f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 4x - 5

Answer

x² - 4x - 5

Question 16

Determine the equation of the tangent line to the graph of f(x) = e^x at (0, 1).

Accepted Answer

y = x + 1

Answer

y = 2x + 1

Answer

y = x

Answer

y = 1

Question 17

Describe the relationship between the derivative of a function and its graph.

Accepted Answer

The derivative represents the slope of the graph

Answer

The derivative represents the area under the graph

Answer

The derivative represents the concavity of the graph

Answer

The derivative represents the maximum or minimum value of the graph

Question 18

Which function exhibits a constant derivative?

Accepted Answer

f(x) = e^x

Answer

f(x) = x²

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 19

Calculate the derivative of f(x) = ln(x² + 1).

Accepted Answer

2x/(x² + 1)

Answer

(2x)/(x² - 1)

Answer

1/(x² + 1)

Answer

x/(x² + 1)

Question 20

Find the local maximum value of the function f(x) = -x^2 + 6x - 5.

Accepted Answer

3

Answer

1

Answer

5

Answer

7

Question 21

Find the derivative of the function f(x) = log(x) using the chain rule.

Accepted Answer

1/x

Answer

x

Answer

log(x)

Answer

1

Question 22

Find the derivative of the function f(x) = x³ - 2x² + 5, clearly indicating the steps and reasons you are using to find the answer.

Accepted Answer

3x² - 4x

Answer

3x² + 4x

Answer

3x - 4

Answer

3x + 4

Question 23

What is the slope of the tangent line to the curve y = sin(x) at x = π/4?

Accepted Answer

√2

Answer

1

Answer

0

Answer

-1

Question 24

Find the derivative of the function f(x) = e^x + ln(x), applying the relevant rules of differentiation.

Accepted Answer

e^x + 1/x

Answer

e^x - 1/x

Answer

e^x / x

Answer

(e^x + ln(x)) * x

Question 25

What is the derivative of the function f(x) = √(x² + 1)?

Accepted Answer

x / √(x² + 1)

Answer

1 / √(x² + 1)

Answer

2x / √(x² + 1)

Answer

(x² + 1) / 2x √(x² + 1)

Question 26

Use the chain rule to calculate the derivative of g(x) = sin(x).

Accepted Answer

cos(x)

Answer

sin(x)

Answer

1

Answer

0

Question 27

Apply the first derivative test to determine where f(x) = x^4 - 2x^2 is increasing or decreasing.

Accepted Answer

Increasing on (-∞, -1) and (1, ∞), decreasing on (-1, 1)

Answer

Increasing on (-1, 1), decreasing on (-∞, -1) and (1, ∞)

Answer

Decreasing on the entire real line

Answer

Increasing on the entire real line

Question 28

Find the derivative of the function h(x) = sin(x) + cos(x).

Accepted Answer

cos(x) + sin(x)

Answer

cos(x) - sin(x)

Answer

-sin(x) - cos(x)

Answer

sin(x) - cos(x)

Question 29

Find the derivative of the function f(x) = e^(2x - 1).

Accepted Answer

2e^(2x - 1)

Answer

e^(2x - 1)

Answer

2xe^(2x - 1)

Answer

xe^(2x - 1)

Question 30

Calculate the derivative of the function g(x) = ln(x² + 1).

Accepted Answer

2x / (x² + 1)

Answer

1 / (x² + 1)

Answer

2x

Answer

(x² + 1) / x

Question 31

Calculate the derivative of the function k(x) = (x³ - 2x) / (x² + 1).

Accepted Answer

(2x³ - 4x² - 2) / (x² + 1)²

Answer

(x³ - 2x)² / (x² + 1)

Answer

(x³ - 4x) / (x² + 1)

Answer

(2x³ - 2x) / (x² + 1)

Question 32

Find the derivative of the function f(x) = sin(x²) + cos(x³).

Accepted Answer

2x cos(x²) + 3x² sin(x³)

Answer

cos(x²) - sin(x³)

Answer

sin(x²) + cos(x³)

Answer

2x sin(x²) - 3x² cos(x³)

Question 33

Calculate the derivative of the function g(x) = e^(sin(x)) + ln(cos(x)).

Accepted Answer

e^(sin(x)) cos(x) + sin(x) / cos(x)

Answer

e^(sin(x)) cos(x) - sin(x) / cos(x)

Answer

e^(sin(x)) - ln(cos(x))

Answer

e^(sin(x)) + ln(cos(x))

Question 34

Which of the following statements accurately defines the derivative of a function f(x)?

Accepted Answer

The limit of the slope of the secant lines connecting two points on the function as the distance between those points approaches zero

Answer

The average rate of change of the function over a specified interval

Answer

The area under the curve of the function

Question 35

Find the derivative of the polynomial function f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x - 5

Answer

3x³ + 4x² - 5

Answer

x² + 4x - 5

Question 36

Find the derivative of the exponential function f(x) = e^x.

Accepted Answer

e^x

Answer

e^(x-1)

Answer

1/e^x

Answer

x * e^x

Question 37

Find the derivative of the radical function f(x) = √x.

Accepted Answer

1/(2√x)

Answer

x^(1/2)

Answer

√x

Answer

1/√x

Question 38

Find the derivative of the composite function f(x) = sin(x² + 1).

Accepted Answer

2xcos(x² + 1)

Answer

cos(x² + 1)

Answer

2xsin(x² + 1)

Answer

sin(x² + 1)

Question 39

For the graph of f(x) = sin(x), what is the slope of the tangent line at x = π/4?

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

2

Question 40

Calculate the derivative of the function f(x) = x^2 + 2x - 1.

Accepted Answer

2x + 2

Answer

x^2 + 2

Answer

1

Answer

2x - 1

Question 41

Determine the derivative of the function f(x) = sin(x).

Accepted Answer

cos(x)

Answer

xcos(x)

Answer

-sin(x)

Answer

sin(x)

Question 42

Find the derivative of the function f(x) = e^x.

Accepted Answer

e^x

Answer

x^e

Answer

e

Answer

1

Question 43

What is the derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

x

Answer

-1/x

Answer

ln(x)

Question 44

Find the derivative of the function f(x) = sqrt(x).

Accepted Answer

1/(2sqrt(x))

Answer

1/sqrt(x)

Answer

2sqrt(x)

Question 45

What is the derivative of the function f(x) = arctan(x)?

Accepted Answer

1/(1+x^2)

Answer

x

Answer

arctan(x)

Answer

-1/(1+x^2)

Question 46

Calculate the derivative of the function f(x) = x^2 + 2x - 3 using the power rule.

Accepted Answer

f'(x) = 2x + 2

Answer

f'(x) = x + 2

Answer

f'(x) = x - 3

Answer

f'(x) = x^2 + 2

Question 47

Calculate the derivative of the function h(x) = e^(2x - 1) using the chain rule.

Accepted Answer

h'(x) = 2e^(2x - 1)

Answer

h'(x) = e^(2x - 2)

Answer

h'(x) = (2x - 1)*e^(2x - 1)

Answer

h'(x) = 2x*e^(2x - 1)

Question 48

A company's profit function is given by P(x) = -x^2 + 100x - 200, where x represents the number of units sold. To maximize profit, find the value of x where the derivative is equal to zero.

Accepted Answer

x = 50

Answer

x = 25

Answer

x = 100

Answer

x = 75

Question 49

A particle's position function is given by s(t) = t^3 - 6t^2 + 9t. Find the particle's acceleration at time t = 2. Acceleration is the second derivative of position.

Accepted Answer

a(2) = 5 m/s^2

Answer

a(2) = 10 m/s^2

Answer

a(2) = -5 m/s^2

Question 50

Identify the function with a negative definite derivative, which implies that the function is strictly decreasing.

Accepted Answer

f(x) = -x^2

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = |x|