Master Calculus Derivatives: Enhance Your Mathematical Skills

Question 1

Which of these is NOT a technique for finding the derivative of a function?

Accepted Answer

Integration by substitution

Answer

Power rule

Answer

Product rule

Answer

Chain rule

Question 2

Express the derivative of f(x) = x^3 - 2x + 1 using the three common notations: Leibniz, Lagrange, and Newton.

Accepted Answer

Leibniz: dy/dx = 3x^2 - 2

Accepted Answer

Lagrange: f'(x) = 3x^2 - 2

Accepted Answer

Newton: d/dx (x^3 - 2x + 1) = 3x^2 - 2

Question 3

Which of the following correctly represents the derivative of the function f(x) = x³ - 2x² + 5?

Accepted Answer

3x² - 4x

Answer

x³ - 2x

Answer

x² - 2

Question 4

Which of the following represents the derivative of the logarithmic function f(x) = log₂(x)?

Accepted Answer

1/(x ln 2)

Answer

ln x

Answer

2/x

Answer

ln 2

Question 5

Describe the relationship between the first derivative and the concavity of a function.

Accepted Answer

Positive derivative indicates concavity up, negative derivative indicates concavity down.

Answer

Zero derivative indicates concavity up, non-zero derivative indicates concavity down.

Answer

Positive derivative indicates concavity down, negative derivative indicates concavity up.

Question 6

Which of the following is an example of an application of derivatives in economics?

Accepted Answer

Determining the marginal cost of production.

Answer

Calculating the area of a triangle.

Answer

Solving differential equations.

Question 7

Find the derivative of the function f(x) = |x - 2|.

Accepted Answer

1 if x > 2, -1 if x < 2

Answer

1

Answer

0

Answer

-1

Question 8

Determine the derivative of the function f(x) = x³ + 3x² - 2.

Accepted Answer

f'(x) = 3x² + 6x

Answer

f'(x) = 3x + 6

Answer

f'(x) = 1

Answer

f'(x) = x³ + 6x²

Question 9

A company's profit, *P*, in thousands of dollars, from producing and selling *x* units of a product is given by  *P(x) = -0.02x² + 50x - 100*. What is the marginal profit when 1000 units are produced?

Accepted Answer

30 thousand dollars

Answer

50 thousand dollars

Answer

40 thousand dollars

Answer

10 thousand dollars

Question 10

Find the derivative of *f(x) = 3x² + 2x - 1*.

Accepted Answer

f'(x) = 6x + 2

Answer

f'(x) = 9x² + 2

Answer

f'(x) = 3x + 2

Answer

f'(x) = 6x - 1

Question 11

What is the derivative of *sin(x)* with respect to *x*?

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

tan(x)

Answer

-cos(x)

Question 12

If *y = e^x*, then *dy/dx* equals:

Accepted Answer

e^x

Answer

1/x

Answer

ln(x)

Answer

x*e^(x-1)

Question 13

Find the derivative of *y = (x² + 1)(x - 3)*.

Accepted Answer

3x² - 6x + 1

Answer

2x

Answer

x² - 6x + 1

Answer

3x² - 6x - 3

Question 14

What is the derivative of *y = ln(2x + 5)*?

Accepted Answer

2/(2x + 5)

Answer

1/(x + 5)

Answer

2x/(2x + 5)

Answer

1/(2x + 5)

Question 15

What is the second derivative of *y = 4x³ - 5x² + 2x - 1*?

Accepted Answer

24x - 10

Answer

12x² - 10x + 2

Answer

12x - 10

Question 16

A spherical balloon is being inflated at a rate of 5 cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 2 inches?

Accepted Answer

5/(16π) inches per minute

Answer

80π inches per minute

Answer

5/8 inches per minute

Answer

5π/16 inches per minute

Question 17

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x^2 - 5x

Answer

3x^2 + 2x - 5

Answer

x^2 + 4x - 5

Question 18

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

Accepted Answer

cos^2(x) - sin^2(x)

Answer

-sin^2(x) - cos^2(x)

Answer

sin(x)cos(x)

Answer

sin(x) + cos(x)

Question 19

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

Accepted Answer

e^x

Answer

e^(x^2)

Answer

1/e^x

Answer

x * e^x

Question 20

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

Accepted Answer

1/x

Answer

-1/x

Answer

ln(x^2)

Answer

x * ln(x)

Question 21

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

Accepted Answer

1/(2sqrt(x))

Answer

x^(1/2)

Answer

-1/(2sqrt(x))

Answer

sqrt(x)

Question 22

What is the derivative of f(x) = 1/x?

Accepted Answer

-1/x^2

Answer

1/x

Answer

x^2 * 1/x

Answer

x * 1/x

Question 23

Find the derivative of f(x) = |x|.

Accepted Answer

1 if x > 0, -1 if x < 0

Answer

0 if x >= 0, -1 if x < 0

Answer

1 if x >= 0, 0 if x < 0

Answer

x if x >= 0, -x if x < 0

Question 24

What is the derivative of f(x) = x^2 * sin(x)?

Accepted Answer

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

Answer

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

Answer

x^2 * cos(x) - sin(x)

Answer

sin(x) + x^2

Question 25

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

Accepted Answer

(x^4 - 2x^3 + 4x + 2)/(x^2 - 1)^2

Answer

(x^4 - 2x^3 + 4x - 2)/(x^2 - 1)

Answer

(x^3 + 2x)/(x^2 - 1)^2

Answer

(x^4 + 2x^3 + 4x + 2)/(x^2 - 1)^2

Question 26

What is the derivative of f(x) = f(g(x))?

Accepted Answer

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

Answer

f'(x) * g'(x)

Answer

f'(g(x)) / g'(x)

Question 27

What is the derivative of f(x) = 3x² + 2x - 1?

Accepted Answer

f'(x) = 6x + 2

Answer

f'(x) = 9x + 2

Answer

f'(x) = 3x + 2

Answer

f'(x) = 6x² + 2x

Question 28

Find the derivative of g(x) = sin(x) at x = π/2.

Accepted Answer

g'(π/2) = 0

Answer

g'(π/2) = -1

Answer

g'(π/2) = π/2

Answer

g'(π/2) = 1

Question 29

The position of a particle is given by s(t) = t³ - 6t² + 9t. What is the particle's velocity at t = 2?

Accepted Answer

v(2) = -3

Answer

v(2) = -9

Answer

v(2) = 9

Answer

v(2) = 3

Question 30

Find the critical points of the function h(x) = x⁴ - 4x³ + 4x².

Accepted Answer

x = 0, x = 1, x = 2

Answer

x = 0, x = 1

Answer

x = 1, x = 2

Answer

x = 0, x = 2

Question 31

The slope of the tangent line to the curve y = x² - 3x at the point (2, -2) is:

Accepted Answer

1

Answer

-1

Answer

2

Question 32

What is the derivative of the function f(x) = e^(2x) ?

Accepted Answer

f'(x) = 2e^(2x)

Answer

f'(x) = 2e^x

Answer

f'(x) = e^(x^2)

Answer

f'(x) = e^(2x)

Question 33

Find the derivative of y = (x² + 1)/(x - 1).

Accepted Answer

y' = (x² - 2x - 1)/(x - 1)²

Answer

y' = (x² - 2x - 1)/(x + 1)²

Answer

y' = (x² + 2x + 1)/(x - 1)²

Answer

y' = (x² + 2x + 1)/(x + 1)²

Question 34

The function f(x) = x³ - 3x² + 3x - 1 has a local maximum at:

Accepted Answer

x = 1

Answer

x = -1

Answer

x = 0

Answer

x = 2

Question 35

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

Accepted Answer

f''(x) = -1/x²

Answer

f''(x) = -1/x

Answer

f''(x) = 1/x²

Answer

f''(x) = 1/x

Question 36

A company's profit function is given by P(x) = -x³ + 6x² - 9x, where x is the number of units produced. How many units should the company produce to maximize its profit?

Accepted Answer

x = 3

Answer

x = 1

Answer

x = 4

Answer

x = 2

Question 37

What is the derivative of the function f(x) = 3x² - 5x + 2?

Accepted Answer

6x - 5

Answer

9x - 5

Answer

3x - 5

Answer

6x² - 5x

Question 38

The position of a particle moving along a straight line is given by s(t) = 2t³ - 9t² + 12t - 4, where t is time in seconds.  What is the velocity of the particle at t = 2 seconds?

Accepted Answer

0

Answer

-6

Answer

12

Answer

4

Question 39

Find the derivative of f(x) = (x² + 1)(x - 3).

Accepted Answer

3x² - 6x + 1

Answer

x² - 6x + 1

Answer

2x² - 6x + 1

Answer

3x² - 6x - 3

Question 40

What is the derivative of the function g(x) = sin(2x)?

Accepted Answer

2cos(2x)

Answer

-2cos(2x)

Answer

2sin(2x)

Question 41

The height of a projectile launched vertically upwards is given by h(t) = -5t² + 40t + 20, where t is time in seconds. What is the maximum height attained by the projectile?

Accepted Answer

100

Answer

80

Answer

40

Answer

20

Question 42

What is the derivative of the function h(x) = e^(3x + 1)?

Accepted Answer

3e^(3x + 1)

Answer

(3x + 1)e^(3x)

Answer

e^(3x + 1)

Question 43

Find the equation of the tangent line to the curve y = x³ - 2x + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = 3x - 3

Answer

y = -3x + 3

Answer

y = -x + 1

Question 44

A spherical balloon is being inflated at a rate of 50 cubic centimeters per second.  How fast is the radius of the balloon increasing when the radius is 10 centimeters? (Volume of a sphere: V = (4/3)πr³)

Accepted Answer

1/(8π) cm/s

Answer

10π cm/s

Answer

5/(4π) cm/s

Question 45

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

Accepted Answer

(2x)/(x² + 1)

Answer

(2x)(x² + 1)

Answer

2xln(x² + 1)

Answer

1/(x² + 1)

Question 46

Find the critical points of the function f(x) = x³ - 6x² + 9x.

Accepted Answer

x = 1, x = 3

Answer

x = 0, x = 6

Answer

x = 0, x = 2

Answer

x = -1, x = -3