Master Derivatives: The Ultimate Guide to Finding Slopes of Functions

Question 1

Find the derivative of the function f(x) given by f(x) = (x² + 3)(x - 4).

Accepted Answer

2x³ - 11x² + 12x

Answer

2x³ + 11x² - 12x

Answer

x³ - 11x² + 12x

Answer

x³ + 11x² - 12x

Question 2

Find the derivative of the function g(x) given by g(x) = tan(3x - 2).

Accepted Answer

3sec²(3x - 2)

Answer

sec²(3x - 2)

Answer

3tan(3x - 2)

Answer

tan(3x - 2)

Question 3

Calculate the derivative of the function h(x) given by h(x) = cot(x²) + csc(x).

Accepted Answer

-2x csc²(x²) + csc(x) cot(x)

Answer

2x csc²(x²) + csc(x) cot(x)

Answer

-2x sec²(x²) + sec(x) tan(x)

Answer

2x sec²(x²) + sec(x) tan(x)

Question 4

Calculate the derivative of the function f(x) = (x^2 + 3x - 5)^4.

Accepted Answer

4(x^2 + 3x - 5)^3(2x + 3)

Answer

4(x + 3)(x^2 + 3x - 5)^3

Answer

4x(x^2 + 3x - 5)^3

Answer

(x^2 + 3x - 5)^4

Question 5

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 6

Which method should be used to calculate the derivative of f(x) = sin(x^2)?

Accepted Answer

Chain rule

Answer

Power rule

Answer

Product rule

Answer

Exponential rule

Question 7

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2xe^(2x)

Answer

e^(x^2)

Question 8

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

Accepted Answer

3x^2/x

Answer

x^2/3x

Answer

3x/x^2

Answer

1/3x^2

Question 9

Which of the following is the derivative of f(x) = e^x?

Accepted Answer

e^x

Answer

x * e^x

Answer

1 / e^x

Answer

ln(e^x)

Question 10

Find the derivative of g(x) = ln(x^2 + 1).

Accepted Answer

2x / (x^2 + 1)

Answer

1 / (x^2 + 1)

Answer

2x * ln(x^2 + 1)

Answer

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

Question 11

Which of the following is NOT a rule for finding derivatives?

Accepted Answer

Quotient rule

Answer

Power rule

Answer

Product rule

Answer

Chain rule

Question 12

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

Accepted Answer

12x^2 - 6

Answer

12x^2 + 6

Answer

4x^3 - 6x + 2

Answer

4x^3 + 6x - 2

Question 13

Which of the following functions has a constant derivative?

Accepted Answer

e^x

Answer

x^2

Answer

sin(x)

Answer

ln(x)

Question 14

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

Accepted Answer

3x² - 4x

Answer

3x² - 2x

Answer

x² - 4x

Answer

x² - 2x

Question 15

Apply the product rule to find the derivative of g(x) = (x + 2)(x - 1).

Accepted Answer

x² + x - 2

Answer

2x - 1

Answer

2x + 1

Answer

x² - x - 2

Question 16

Find the derivative of f(x) = e^(3x) using the chain rule.

Accepted Answer

3e^(3x)

Answer

e^(3x)

Answer

3x^(e)

Answer

3e^(x)

Question 17

Find the derivative of f(x) = (x² + 1)/(x - 1) using the quotient rule.

Accepted Answer

(2x - 1)/(x-1)²

Answer

(2x + 1)/(x-1)²

Answer

(x² - 1)/(x-1)²

Answer

(x² + 2)/(x-1)²

Question 18

Find the local maximum of the function f(x) = x² - 4x + 3.

Accepted Answer

x = 2

Answer

x = 0

Answer

x = -2

Answer

The function does not have a local maximum.

Question 19

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

Accepted Answer

6x - 2

Answer

2x² - 2

Answer

3x² - 2x

Answer

6x + 2

Question 20

Find the derivative of the function y = x^3 + 2x^2 - 5x + 1 using the power rule.

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 - 4x + 5

Answer

x^3 + 4x^2 - 5

Answer

x^3 - 2x^2 - 5

Question 21

Find the derivative of g(x) = sin(2x) using the chain rule.

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-sin(2x)

Answer

-2sin(2x)

Question 22

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

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = -2

Answer

x = 2

Answer

x = -2

Question 23

Which of the following statements is correct about the derivative of a constant function?

Accepted Answer

The derivative is always 0.

Answer

The derivative is always a constant.

Answer

The derivative is always the original function.

Answer

The derivative does not exist.

Question 24

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

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Answer

-sin(2x)

Question 25

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

Accepted Answer

2x / (x² + 1)

Answer

-2x / (x² + 1)

Answer

1 / (x² + 1)

Answer

-1 / (x² + 1)

Question 26

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

Accepted Answer

sec(x)tan(x)

Answer

-sec(x)tan(x)

Answer

cosec(x)

Answer

-cosec(x)

Question 27

Find the derivative of the function h(x) = e^(3x - 2).

Accepted Answer

h'(x) = 3e^(3x - 2)

Answer

h'(x) = e^(3x - 2)

Answer

h'(x) = 3xe^(3x - 2)

Answer

h'(x) = xe^(3x - 2)

Question 28

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

Accepted Answer

f'(x) = 2x/(x² + 1)

Answer

f'(x) = 1/(x² + 1)

Answer

f'(x) = 2/(x² + 1)

Answer

f'(x) = x/(x² + 1)

Question 29

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

Accepted Answer

g'(x) = (2x(x - 1) - (x² + 1))/(x - 1)²

Answer

g'(x) = (2x(x - 1) + (x² + 1))/(x - 1)²

Answer

g'(x) = (2x - 1)/(x - 1)

Answer

g'(x) = (2x + 1)/(x - 1)

Question 30

Calculate the derivative of g(x) = ln(x³ + 2x).

Accepted Answer

g'(x) = (3x² + 2)/(x³ + 2x)

Answer

g'(x) = (3x² + 2)/(x³ + 2)

Answer

g'(x) = (3x² + 2)

Answer

g'(x) = (3x² + 2x)

Question 31

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

Accepted Answer

g'(x) = x/√(x² + 1)

Answer

g'(x) = √(x² + 1)

Answer

g'(x) = x

Answer

g'(x) = 1

Question 32

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 2x - 5

Answer

3x + 2

Answer

x + 1

Question 33

Determine the derivative of the function f(x) = sin(x^2 + 1).

Accepted Answer

2x * cos(x^2 + 1)

Answer

cos(x^2 + 1)

Answer

sin(x^2 + 1)

Answer

x * cos(x^2 + 1)

Question 34

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

Accepted Answer

2e^(2x + 3)

Answer

e^(2x + 3)

Answer

2x * e^(2x + 3)

Answer

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

Question 35

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 36

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

Accepted Answer

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

Answer

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

Answer

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

Answer

0

Question 37

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

Accepted Answer

2x / (1 + x^4)

Answer

x / (1 + x^4)

Answer

1 / (1 + x^4)

Answer

2 / (1 + x^4)

Question 38

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

Accepted Answer

f'(x) = 3x^2 - 4x + 5

Answer

f'(x) = x^3 - 2x + 5

Answer

f'(x) = 3x^3 - 4x^2 + 5x - 1

Answer

f'(x) = x^2 - 4x + 5

Question 39

Find the derivative of the function h(x) = sin(2x + 1).

Accepted Answer

h'(x) = 2cos(2x + 1)

Answer

h'(x) = sin(4x + 2)

Answer

h'(x) = cos(2x + 1)

Answer

h'(x) = -2sin(2x + 1)

Question 40

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

Accepted Answer

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

Answer

f'(x) = 2x

Answer

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

Answer

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

Question 41

Find the derivative of the function g(x) = ln(x^2 + 5).

Accepted Answer

g'(x) = 2x/(x^2 + 5)

Answer

g'(x) = 2x

Answer

g'(x) = 1/(x^2 + 5)

Answer

g'(x) = ln(2x)

Question 42

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

Accepted Answer

h'(x) = 1/(1 + x^2)

Answer

h'(x) = 1/x

Answer

h'(x) = x/(1 + x^2)

Answer

h'(x) = arctan(1 + x)

Question 43

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

Accepted Answer

f'(x) = (2x)(x - 3) + (x² + 1)(1)

Answer

f'(x) = (2x)(x - 3) - (x² + 1)(1)

Answer

f'(x) = (x² + 3)(x - 1)

Answer

f'(x) = (x² + 1)(x - 2)

Question 44

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

Accepted Answer

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

Answer

f'(x) = x*cos(x²)

Answer

f'(x) = -sin(x²)

Answer

f'(x) = cos(x²)

Question 45

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 46

Find the derivative of the function f(x) = ln(x² + 3).

Accepted Answer

f'(x) = 2x/(x² + 3)

Answer

f'(x) = 2x

Answer

f'(x) = 1/(x² + 3)

Answer

f'(x) = ln(x² + 3)

Question 47

Find the derivative of the function f(x) = arctan(x³).

Accepted Answer

f'(x) = 3x²/(1 + x^6)

Answer

f'(x) = arctan(x³)

Answer

f'(x) = 3x²

Answer

f'(x) = 1/(1 + x^6)