Master Differentiation Rules: Enhance Your Calculus Skills

Question 1

What is the derivative of y = sin(x)?

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-cos(x)

Answer

-sin(x)

Question 2

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

Accepted Answer

e^x

Answer

x*e^x

Answer

1

Answer

e^-x

Question 3

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

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

-1/x

Question 4

What is the derivative of y = x^3 + 2x^2 - 5x + 1?

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 - 4x + 5

Answer

x^2 + 2x - 5

Answer

x^3 - 4x^2 + 10x - 5

Question 5

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

Accepted Answer

2xe^(x^2)

Answer

x^2e^(x^2)

Answer

e^(x^2)

Answer

xe^(x^2)

Question 6

Find the derivative of f(x) = 2x³ + 5x² - 3x + 1 using the power rule.

Accepted Answer

6x² + 10x - 3

Answer

2x² + 5x - 3

Answer

6x + 10

Answer

2x + 5

Question 7

Calculate the rate of change of the function f(x) = sin(2x) at x = π/4.

Accepted Answer

√2

Answer

0

Answer

1

Answer

2

Question 8

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

Accepted Answer

g'(x) = 2e^(2x + 1)

Answer

g'(x) = e^(2x + 1)

Answer

g'(x) = 2e^2x

Answer

g'(x) = e^2x

Question 9

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

Accepted Answer

h'(x) = 2x/(x² + 1)

Answer

h'(x) = (2x)/(x² - 1)

Answer

h'(x) = x/(x² + 1)

Answer

h'(x) = 1/(x² + 1)

Question 10

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

Accepted Answer

f'(x) = (4x² - 9x + 3)/(4x² - 4x + 1)

Answer

f'(x) = (4x² - 9x - 3)/(4x² - 4x + 1)

Answer

f'(x) = (4x² + 9x + 3)/(4x² - 4x + 1)

Answer

f'(x) = (4x² + 9x - 3)/(4x² - 4x + 1)

Question 11

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

Accepted Answer

h'(x) = 2x cos(x² + 1)

Answer

h'(x) = cos(x² + 1)

Answer

h'(x) = 2x sin(x² + 1)

Answer

h'(x) = sin(x² + 1)

Question 12

Find the derivative of the function g(x) = √(x³ + 2x - 1).

Accepted Answer

g'(x) = (3x² + 2) / (2√(x³ + 2x - 1))

Answer

g'(x) = (3x² - 2) / (2√(x³ + 2x - 1))

Answer

g'(x) = (x² + 2) / (2√(x³ + 2x - 1))

Answer

g'(x) = (x² - 2) / (2√(x³ + 2x - 1))

Question 13

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

Accepted Answer

2x

Answer

2x - 2

Answer

2x + 2

Answer

x^2

Question 14

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

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-cos(x)

Answer

1

Question 15

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

Accepted Answer

1/x

Answer

1

Answer

ln(x)

Answer

x

Question 16

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

Accepted Answer

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

Answer

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

Answer

2x * (2x - 3)

Answer

(x^2 + 1) * 1

Question 17

Determine the derivative of f(x) = x^3 - 2x^2 + e^x.

Accepted Answer

3x^2 - 4x + e^x

Answer

x^3 - 2x

Answer

3x^2 - 4x

Answer

e^x

Question 18

Determine the derivative of f(x) = (x^2 + 1)^3 using the chain rule.

Accepted Answer

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

Answer

(x^2 + 1)^3

Answer

3

Answer

2x

Question 19

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

Accepted Answer

3x^2 - 4x + 5

Answer

6x^2 - 4x + 5

Answer

3x - 4

Answer

6x - 4

Question 20

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

Accepted Answer

e^x

Answer

e

Answer

x

Answer

1

Question 21

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

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

1

Question 22

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

Accepted Answer

cos(x)

Answer

sin(x)

Answer

x

Answer

1

Question 23

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

Accepted Answer

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

Answer

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

Answer

(x + 1) / (x - 1)

Answer

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

Question 24

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

Accepted Answer

sec^2(x)

Answer

1 / sec^2(x)

Answer

cos(x) / sin(x)

Answer

sin(x) / cos(x)

Question 25

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

Accepted Answer

3x^2 - 2

Answer

x^2 - 2

Answer

2x^2 - 1

Answer

x^2 + 1

Question 26

Determine the derivative of the function y = sin(x) + cos(x).

Accepted Answer

cos(x) + sin(x)

Answer

sin(x) - cos(x)

Answer

cos(x) - sin(x)

Answer

sin(x) + cos(x)

Question 27

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

Accepted Answer

x * cos(x) + sin(x)

Answer

cos(x) - x * sin(x)

Answer

sin(x) - x * cos(x)

Answer

cos(x) + x * sin(x)

Question 28

Determine the derivative of the function y = (x^2 + 1)/(x - 1).

Accepted Answer

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

Answer

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

Answer

2x/(x - 1)

Answer

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

Question 29

Determine the derivative of the function y = sqrt(x^2 - 4).

Accepted Answer

x/(2 * sqrt(x^2 - 4))

Answer

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

Answer

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

Answer

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

Question 30

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

Accepted Answer

1/(1 + x^2)

Answer

(1 + x^2)/x

Answer

x/(1 + x^2)

Answer

(1 - x^2)/x

Question 31

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

Accepted Answer

6x + 2

Answer

6x² + 2

Answer

3x + 2

Answer

9x² + 2x

Question 32

Find the derivative of *g(x) = 4x³ - 5x + 7*.

Accepted Answer

12x² - 5

Answer

12x² - 5x

Answer

12x³ - 5

Answer

4x² - 5

Question 33

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

Accepted Answer

6x² + 2x - 3

Answer

2x² + 2x - 3

Answer

4x³ + 2x² - 6x - 3

Answer

6x² - 3

Question 34

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

Accepted Answer

(x² - 2x - 1) / (x - 1)²

Answer

(x² + 2x + 1) / (x - 1)²

Answer

(x² - 2x + 1) / (x - 1)²

Answer

(2x - 1) / (x - 1)²

Question 35

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

Accepted Answer

cos(x) - sin(x)

Answer

-cos(x) - sin(x)

Answer

-cos(x) + sin(x)

Answer

sin(x) + cos(x)

Question 36

Find the derivative of *g(x) = e^x + ln(x)*.

Accepted Answer

e^x + 1/x

Answer

e^x - 1/x

Answer

e^x + x

Answer

e^x + ln(x)

Question 37

What is the derivative of *y = 2^x*?

Accepted Answer

2^x * ln(2)

Answer

2^x

Answer

x * 2^(x-1)

Question 38

Find the derivative of *h(x) = ln(3x² - 2)*.

Accepted Answer

6x / (3x² - 2)

Answer

6x / (3x² + 2)

Answer

1 / (3x² - 2)

Question 39

What is the derivative of *f(x) = (x³ + 2x)²*?

Accepted Answer

2(x³ + 2x)(3x² + 2)

Answer

2(3x² + 2)

Answer

(x³ + 2x)(3x² + 2)

Answer

(x³ + 2x)²(3x² + 2)

Question 40

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

Accepted Answer

3x^2 - 4x + 5

Answer

x^2 - 2x + 5

Answer

3x^2 - 2x + 5

Answer

x^3 - 4x^2 + 5x - 1

Question 41

Using the chain rule, find the derivative of the function  g(x) = e^(2x+1).

Accepted Answer

2e^(2x+1)

Answer

e^(2x+1)

Answer

1 / e^(2x+1)

Answer

2x * e^(2x+1)

Question 42

Apply the chain rule to find the derivative of h(x) = ln(x^2 + 1).

Accepted Answer

2x / (x^2 + 1)

Answer

2x

Answer

ln(2x)

Answer

1 / (x^2 + 1)

Question 43

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

Accepted Answer

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

Answer

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

Answer

e^(-x) + 2x

Answer

-e^(-x) + 2x

Question 44

Employ the product rule to find the derivative of g(x) = x * ln(x).

Accepted Answer

ln(x) + 1

Answer

x * (1 / ln(x))

Answer

x / ln(x)

Answer

1

Question 45

Calculate the derivative of h(x) = sqrt(x^3 + 1) using the chain rule.

Accepted Answer

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

Answer

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

Answer

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

Answer

3x^2 / 2

Question 46

Find the derivative of f(x) = sin(2x) * cos(x) using the product rule.

Accepted Answer

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

Answer

2cos(2x) + 2sin(x)

Answer

sin(2x) + cos(x)

Answer

-sin(2x) * cos(x)

Question 47

Calculate the derivative of g(x) = tan(x^2 - 1) using the chain rule.

Accepted Answer

2x * sec^2(x^2 - 1)

Answer

x * sec^2(x^2 - 1)

Answer

2x * tan(x^2 - 1)

Answer

sec^2(x^2 - 1)

Question 48

Using the chain rule, find the derivative of h(x) = e^(sin(x)).

Accepted Answer

e^(sin(x)) * cos(x)

Answer

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

Answer

e^(cos(x))

Answer

e^(sin(x))

Question 49

What is the derivative of f(x) = ln(x^2 + y^2) with respect to x, assuming y is a constant?

Accepted Answer

2x / (x^2 + y^2)

Answer

2y / (x^2 + y^2)

Answer

y / x

Answer

1 / (x^2 + y^2)

Question 50

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

Accepted Answer

6x + 2

Answer

3x + 2

Answer

6x² + 2x

Question 51

Find the derivative of the function *f(x) = 5x⁴ - 2x³ + 7x*.

Accepted Answer

20x³ - 6x² + 7

Answer

5x³ - 6x² + 7

Answer

20x⁴ - 6x³ + 7

Question 52

Find the derivative of the function *f(x) = 4x³ * 2x²*.

Accepted Answer

32x⁴

Answer

8x⁵

Answer

24x⁵

Answer

12x⁵

Question 53

What is the derivative of the function *f(x) = (x² + 1) / (x - 1)*?

Accepted Answer

(x² - 2x - 1) / (x - 1)²

Answer

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

Answer

(x² - 2x + 1) / (x - 1)²

Answer

(x² + 2x - 1) / (x - 1)²

Question 54

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

Accepted Answer

cos²(x) - sin²(x)

Answer

sin(x) + cos(x)

Answer

cos(x) - sin(x)

Answer

sin²(x) + cos²(x)

Question 55

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

Accepted Answer

2e^(2x)

Answer

e^(x²)

Answer

2e^x

Answer

e^(2x)

Question 56

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

Accepted Answer

2x / (x² + 1)

Answer

2 / (x² + 1)

Answer

1 / (x² + 1)

Answer

x / (x² + 1)

Question 57

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

Accepted Answer

sec²(x)

Answer

csc²(x)

Answer

cot²(x)

Answer

cos²(x)

Question 58

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

Accepted Answer

2^x * ln(2)

Answer

2^(x-1)

Answer

x * 2^(x-1)

Answer

2^x

Question 59

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)