Master Calculus Derivatives: Enhance Your Mathematical Skills

Question 1

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

Accepted Answer

x^3 * e^x + 3x^2 * e^x

Answer

3x^2 * e^x

Answer

x^3 + e^x

Answer

(x^4 + e^x)'

Question 2

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 3

Consider the function f(x) = |x-2| + |x+2|. Determine its continuity and differentiability at x = 2. Explain your reasoning.

Accepted Answer

Continuous at x = 2 but not differentiable at x = 2

Answer

Continuous and differentiable at x = 2

Answer

Neither continuous nor differentiable at x = 2

Question 4

Which of the following is NOT a standard method for finding derivatives?

Accepted Answer

Integration by substitution

Answer

Power rule

Answer

Product rule

Answer

Chain rule

Question 5

Find the derivative of the trigonometric function f(x) = sin(x).

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

xcos(x)

Answer

-xsin(x)

Question 6

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

Accepted Answer

6x(x^2 + 1)^2

Answer

3x(x^2 + 1)^2

Answer

6x^2(x^2 + 1)^2

Answer

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

Question 7

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

Accepted Answer

e^x

Answer

x^e

Answer

e

Answer

x

Question 8

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

Accepted Answer

0, 1, 2

Answer

0, 1

Answer

0, 2

Answer

1, 2

Question 9

Find the derivative of f(x) = x^2 + 3x - 4.

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

2x - 1

Answer

x^2

Question 10

Calculate the derivative of g(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(2x)

Answer

2sin(2x)

Question 11

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

Accepted Answer

e^x

Answer

x*e^x

Answer

1/e^x

Answer

0

Question 12

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

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

ln(x^2 + 1)

Answer

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

Question 13

Find the critical points of f(x) = (x - 1)^2 - 4.

Accepted Answer

x = 1

Answer

x = -3

Answer

x = 3

Answer

x = 5

Question 14

Which derivative application is used to optimize a business's profit function?

Accepted Answer

Finding critical points

Answer

Calculating the slope of a curve

Answer

Determining the second derivative

Answer

Computing the integral of a function

Question 15

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

Accepted Answer

2

Answer

4x

Answer

0

Answer

6x

Question 16

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

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

e^x

Question 17

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

Accepted Answer

e^x

Answer

x

Answer

x*e^x

Answer

1/e^x

Question 18

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Answer

cos(x)/sin(x)

Question 19

What is the derivative of the function f(x) = (x^2 + 1)/(x - 1)?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 20

Calculate 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 21

Calculate the derivative of the function g(x) = arctan(x^2 + 2x).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 22

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

Accepted Answer

sin(x)

Answer

-cos(x)

Answer

sec(x)

Answer

csc(x)

Question 23

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2e^x

Answer

e^x

Question 24

State the Chain Rule for finding the derivative of a composite function.

Accepted Answer

d/dx(f(g(x))) = f'(g(x)) * g'(x)

Answer

d/dx(f(g(x))) = f'(x) * g'(x)

Answer

d/dx(f(g(x))) = g'(f(x)) * f'(x)

Answer

d/dx(f(g(x))) = f'(g'(x))

Question 25

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

Accepted Answer

3(x^2 + 1)^2 * 2x

Answer

(x^2 + 1)^3

Answer

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

Answer

6x(x^2 + 1)^2

Question 26

What is the second derivative of the function f(x) = x^4?

Accepted Answer

24x^2

Answer

12x^2

Answer

6x

Answer

4x

Question 27

What is the derivative of the function f(x) = |2x - 1|?

Accepted Answer

2x - 1 if x >= 1/2, -2x + 1 if x < 1/2

Answer

1

Answer

2

Answer

|2x - 1|

Question 28

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

Accepted Answer

2x

Answer

1/x

Answer

x^2

Answer

2

Question 29

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

-cos(x)

Answer

sin(x)

Question 30

Which of the following is the derivative of the function f(x) = 3x^2 + 2x - 1?

Accepted Answer

6x + 2

Answer

9x + 2

Answer

3x + 2

Answer

6x^2 + 2x

Question 31

Find the derivative of y = sin(x).

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-sin(x)

Answer

-cos(x)

Question 32

Which of the following best describes the instantaneous rate of change of a function?

Accepted Answer

The derivative of the function

Answer

The minimum value of the function

Answer

The maximum value of the function

Answer

The average rate of change of the function

Question 33

If f(x) = x^3 + 2x^2 - 5x, what is the slope of the tangent line to the curve at x = 1?

Accepted Answer

6

Answer

8

Answer

3

Answer

-1

Question 34

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

Accepted Answer

e^x

Answer

1/(e^x)

Answer

ln(x)

Answer

x*e^(x-1)

Question 35

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

Accepted Answer

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

Answer

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

Answer

(2x - 1)/(x - 2)

Answer

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

Question 36

Find the second derivative of f(x) = x^4 + 3x^2.

Accepted Answer

12x^2 + 6

Answer

12x^3 + 6x

Answer

4x^3 + 6x

Answer

24x

Question 37

A particle moves along a straight line with its position given by s(t) = t^3 - 6t^2 + 9t. At what time(s) is the particle at rest?

Accepted Answer

t = 1 and t = 3

Answer

t = 0 and t = 3

Answer

t = 1 and t = 2

Answer

t = 0 and t = 1

Question 38

If the derivative of f(x) is f'(x) = 2x - 3, what is the original function f(x)?

Accepted Answer

f(x) = x^2 - 3x + C

Answer

f(x) = x^2 + 3x + C

Answer

f(x) = x^2 - 3x

Answer

f(x) = 2x^2 - 3x + C

Question 39

The chain rule is used to find the derivative of:

Accepted Answer

Composite functions

Answer

Polynomial functions

Answer

Trigonometric functions

Answer

Linear functions