Master Derivatives: The Ultimate Guide to Measuring Change

Question 1

Derivatives are essential in practical applications because they:

Accepted Answer

Have versatile applications in various fields.

Answer

Are only used for theoretical mathematical study.

Answer

Find exclusive use in physics and economics.

Question 2

How does the derivative of a function relate to the shape of its graph?

Accepted Answer

The derivative at a point gives the slope of the tangent line to the graph at that point.

Answer

The derivative is the area under the graph of the function over an interval.

Answer

The derivative is the value of the function at the maximum or minimum point.

Answer

The derivative is the difference in the function's values over an interval.

Question 3

What is a critical point of a function, and how does it relate to the derivative?

Accepted Answer

A point where the derivative of the function is zero or undefined.

Answer

A point where the function changes concavity.

Answer

A point where the function is discontinuous.

Answer

A point where the function has the steepest slope.

Question 4

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

Accepted Answer

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

Answer

The slope of the tangent line to the graph of f(x) at any point x.

Answer

The area under the curve of f(x) between two points.

Answer

The maximum value of f(x).

Question 5

Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 + 2x - 5

Answer

x^2 + 2x - 5

Question 6

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

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 7

Which of the following is NOT a step in the process of finding the derivative of a function?

Accepted Answer

Integrate the function.

Answer

Simplify the function.

Answer

Apply the appropriate differentiation rule.

Answer

Check your answer.

Question 8

How do derivatives contribute to finding optimal solutions in optimization problems?

Accepted Answer

They locate the maximum or minimum points of a function.

Answer

They determine the slope of normal lines.

Answer

They predict the future values of a function.

Answer

They solve differential equations.

Question 9

The derivative of a function f(x) at a point x is defined as the slope of the tangent line to the graph of f(x) at that point. Which of the following is the correct formula for the derivative of f(x)?

Accepted Answer

$limlimits_{h	o 0}rac{f(x+h)-f(x)}{h}$

Answer

$rac{df}{dx}$

Answer

$rac{d}{dx}f(x)$

Answer

$int f(x)dx$

Question 10

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

Accepted Answer

$cos(x)$

Answer

$sin(x)$

Answer

$-cos(x)$

Answer

$-sin(x)$

Question 11

Which of the following statements about the derivative of a function is true?

Accepted Answer

The derivative can be positive, negative, or zero.

Answer

The derivative is always positive.

Answer

The derivative is always negative.

Answer

The derivative is always zero.

Question 12

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

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 5x + 1

Answer

x² - 5x + 1

Question 13

Find the derivative of f(x) = e^(3x) + ln(x).

Accepted Answer

3e^(3x) + 1/x

Answer

e^(3x) + 1/x

Answer

3e^(3x) - 1/x

Answer

e^(3x) - 1/x

Question 14

If f(x) is differentiable at x = 0, which of the following must be true?

Accepted Answer

The graph of f(x) is continuous at x = 0.

Answer

The graph of f(x) has a sharp corner at x = 0.

Answer

The graph of f(x) has a vertical tangent at x = 0.

Answer

The graph of f(x) has a horizontal tangent at x = 0.

Question 15

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

Accepted Answer

2x + 3

Answer

x^2 + 3x

Answer

2

Answer

3x - 5

Question 16

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

Accepted Answer

e^x

Answer

x * e^x

Answer

e^(x^2)

Answer

1/x * e^x

Question 17

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

Accepted Answer

1/x

Answer

x

Answer

ln(x) * x

Answer

1/(x^2)

Question 18

If f(x) is a differentiable function, find the derivative of the function g(x) = f(x^2).

Accepted Answer

2x * f'(x)

Answer

2x * f(x)

Answer

f(x) * f'(x)

Answer

f(x^2) * f'(x^2)

Question 19

Find the critical points of the function f(x) = x^3 - 3x^2 + 2x + 1.

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = 1

Answer

x = 1, x = 2

Answer

x = 0

Question 20

Find the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis.

Accepted Answer

4π/5 * (16^5 - 0^5)

Answer

π/5 * (16^5 - 0^5)

Answer

2π/5 * (16^5 - 0^5)

Answer

8π/5 * (16^5 - 0^5)

Question 21

State the chain rule for derivatives:

Accepted Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = h'(g(x)) * g'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) / h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = h'(g(x)) / g'(x)

Question 22

Find the derivative of the implicit function defined by the equation x^3 + y^3 = 6xy:

Accepted Answer

dy/dx = (2x^2 - 3y^2) / (3x^2 - 2y^2)

Answer

dy/dx = (3x^2 - 2y^2) / (2x^2 - 3y^2)

Answer

dy/dx = y/x

Answer

dy/dx = x/y

Question 23

Find the maximum value of the function f(x) = -x^2 + 4x - 3 on the interval [0, 3]:

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 24

Find the derivative of the function f(x) = x² + 2x.

Accepted Answer

2x + 2

Answer

2x

Answer

1

Answer

x + 2

Question 25

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

Accepted Answer

cos(x) + sin(x)

Answer

cos(x) - sin(x)

Answer

2cos(x)

Answer

-cos(x) - sin(x)

Question 26

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

Accepted Answer

e^x

Answer

xe^x

Answer

1

Answer

e^(-x)

Question 27

Calculate the derivative of the function g(x) = (x² + 1)³.

Accepted Answer

3(x² + 1)² * 2x

Answer

3x(x² + 1)²

Answer

(x² + 1)³

Answer

2x(x² + 1)²

Question 28

Find the derivative of the function h(x) = arctan(x).

Accepted Answer

1/(1 + x²)

Answer

-1/(1 + x²)

Answer

arctan(x)

Answer

x/(1 + x²)

Question 29

Calculate the derivative of the function f(x) = 1/x.

Accepted Answer

-1/x²

Answer

1/x²

Answer

x

Answer

-x

Question 30

Find the derivative of the function g(x) = √x.

Accepted Answer

1/2 √x

Answer

-1/2 √x

Answer

2 √x

Answer

-2 √x

Question 31

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

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x² - 5

Answer

3x² + 2x - 5

Answer

x³ + 2x² - 5

Question 32

Find the derivative of the polynomial function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 10

Answer

3x + 4

Question 33

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

Accepted Answer

e^x

Answer

x*e^x

Answer

0

Question 34

Find the derivative of the logarithmic function f(x) = ln(x).

Accepted Answer

1/x

Answer

x

Answer

-1/x

Question 35

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

Accepted Answer

1/(2*sqrt(x))

Answer

1/2*sqrt(x)

Answer

2*sqrt(x)

Question 36

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

Accepted Answer

((x + 1)(2x - 1) - (x^2 + 1))/(x - 1)^2

Answer

((x^2 + 1)(x + 1) - (x - 1))/(x - 1)^2

Answer

((x^2 + 1)(x - 1) + (x - 1))/(x - 1)^2

Question 37

Find the derivative of the composite function f(x) = f(g(x)), where g(x) = x^2 - 1.

Accepted Answer

f'(g(x))*g'(x)

Answer

f(g(x))

Answer

f(g'(x))

Question 38

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

Accepted Answer

2x*cos(x^2)

Answer

cos(x^2)

Answer

sin(2x)

Question 39

Which of the following functions has a positive derivative for all real numbers?

Accepted Answer

f(x) = x² + 1

Answer

f(x) = x³ - 2x

Answer

f(x) = sin(x)

Question 40

The position of a moving object is given by the function s(t) = 2t³ - 3t² + 4t - 1. Find the object's velocity at time t = 2.

Accepted Answer

11

Answer

15

Answer

2

Answer

5

Question 41

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

Accepted Answer

f''(x) = e^x

Answer

f''(x) = e^(-x)

Answer

f''(x) = 2e^x

Answer

f''(x) = xe^x

Question 42

What does the derivative of a function represent at a specific point?

Accepted Answer

The instantaneous rate of change of the function at that point.

Answer

The maximum value of the function.

Answer

The average rate of change of the function over a small interval around that point.

Answer

The area under the curve of the function up to that point.

Question 43

If f(x) = 3x² - 2x + 1, what is the derivative, f'(x)?

Accepted Answer

6x - 2

Answer

6x + 1

Answer

3x - 2

Answer

9x² - 2

Question 44

Which of the following is NOT a valid derivative rule?

Accepted Answer

(f(x) + g(x))' = f'(x)g'(x)

Answer

(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)

Answer

(f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x))/(g(x))²

Answer

(cf(x))' = cf'(x)

Question 45

Find the derivative of y = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(2x)

Answer

-2sin(2x)

Question 46

What is the slope of the line tangent to the curve y = e^x at the point where x = 1?

Accepted Answer

e

Answer

-e

Answer

0

Answer

1

Question 47

What is the derivative of ln(x)?

Accepted Answer

1/x

Answer

1/x²

Answer

ln(x)

Answer

x

Question 48

If the derivative of a function is zero at a particular point, what does this tell us about the function at that point?

Accepted Answer

The function has a critical point, which could be a maximum, minimum, or inflection point.

Answer

The function has an inflection point at that point.

Answer

The function has a maximum at that point.

Answer

The function has a minimum at that point.

Question 49

A rectangular box with a square base and an open top needs to have a volume of 32 cubic meters. What dimensions would minimize the surface area of the box?

Accepted Answer

4 meters by 4 meters by 2 meters

Answer

4 meters by 2 meters by 4 meters

Answer

2 meters by 2 meters by 8 meters

Answer

8 meters by 8 meters by 1/2 meter

Question 50

What is the second derivative of f(x) = x⁴ - 2x³ + 5x - 7?

Accepted Answer

12x² - 12x

Answer

24x - 12

Answer

x⁵/5 - x⁴/2 + 5x²/2 - 7x

Answer

4x³ - 6x² + 5