Calculus Differentiation: Master Derivatives and Applications

Question 1

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

Accepted Answer

3x² - 4x + 5

Answer

x² - 2x + 5

Answer

3x² - 2x - 5

Answer

x³ - 2x² - 5

Question 2

Which of the following is the equation of the normal to the curve y = x² at the point (1, 1)?

Accepted Answer

y = -x + 2

Answer

y = x + 2

Answer

y = -x - 2

Answer

y = x - 2

Question 3

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

Accepted Answer

3e^(3x + 5)

Answer

e^(3x + 5)

Answer

3x²e^(3x + 5)

Answer

xe^(3x + 5)

Question 4

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

Accepted Answer

3x² - 4x + 5

Answer

3x² + 4x + 5

Answer

2x³ - 4x² + 5

Answer

2x³ + 4x² + 5

Question 5

Find the equation of the normal to the curve y = x² + 1 at the point (1, 2).

Accepted Answer

y = -x + 3

Answer

y = x + 1

Answer

y = -2x + 4

Answer

y = 2x

Question 6

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

Accepted Answer

e^x

Answer

x * e^x

Answer

1 / e^x

Answer

e^x / x

Question 7

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

Accepted Answer

-4cos(2x)

Answer

2cos(2x)

Answer

-2sin(2x)

Answer

4sin(2x)

Question 8

Indicate the formula for implicit differentiation.

Accepted Answer

dy/dx = ∂f/∂y / ∂f/∂x

Answer

dy/dx = -∂f/∂x / ∂f/∂y

Answer

dy/dx = ∂f/∂x * ∂f/∂y

Answer

dy/dx = -∂f/∂y * ∂f/∂x

Question 9

Obtain the derivative of the function f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2x e^(2x + 1)

Answer

x e^(2x + 1)

Question 10

Determine the maximum value of the function f(x) = -x^2 + 4x + 3 on the interval [-1, 3].

Accepted Answer

7

Answer

5

Answer

3

Answer

1

Question 11

Which of the following is the derivative of the function f(x) = 2x^3 + 5x^2 - 3x + 1?

Accepted Answer

6x^2 + 10x - 3

Answer

6x^3 + 10x^2 - 3

Answer

2x^2 + 5x - 3

Answer

2x^3 + 5x^2 + 3x + 1

Question 12

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

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Answer

-sin(2x)

Question 13

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

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

2x

Answer

(x^2 + 1)/x

Question 14

Find the second derivative of the function f(x) = x^4 - 3x^2 + 2x - 1.

Accepted Answer

12x^2 - 6x + 2

Answer

12x^3 - 6x^2 + 2

Answer

12x^4 - 6x^2 + 2x

Answer

12x - 6

Question 15

Find the derivative of the function f(x) = x^3 - 2x^2 + 5x - 7.

Accepted Answer

3x^2 - 4x + 5

Answer

x^2 - 4x + 5

Answer

3x - 4

Answer

x - 4

Question 16

Which of the following functions has a derivative that is always negative?

Accepted Answer

f(x) = -x^3

Answer

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

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 17

Find the derivative of f(x) = 2x^3 - 5x^2 + 3x - 7 using the power rule.

Accepted Answer

6x^2 - 10x + 3

Answer

6x^2 + 10x + 3

Answer

6x - 10 + 3

Answer

12x^2 - 5x + 3

Question 18

Calculate the gradient of the tangent to the curve y = sin(x) at x = π/4.

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

-1

Question 19

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

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2xe^(2x + 1)

Answer

xe^(2x + 1)

Question 20

Express the chain rule for differentiation.

Accepted Answer

df/dx = (df/dy) * (dy/dx)

Answer

df/dx = (dy/dx) * (df/dy)

Answer

df/dx = (dy/dx) + (df/dy)

Answer

df/dx = (dx/dy) * (dy/dx)

Question 21

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

Accepted Answer

2x + 2

Answer

2x - 2

Answer

x^2 + 2

Answer

x - 3

Question 22

Identify the inflection point of the curve y = x^3 - 3x^2.

Accepted Answer

(1, -2)

Answer

(0, 0)

Answer

(0, -3)

Answer

(1, 0)

Question 23

Calculate the derivative of f(x) = sin(x)cos(x):

Accepted Answer

cos(x) - sin(x)

Answer

-sin(x)sin(x)

Answer

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

Answer

sin(x) + cos(x)

Question 24

Determine whether the function f(x) = x^3 + 2x^2 - 1 is increasing or decreasing at x = -1:

Accepted Answer

Decreasing

Answer

Increasing

Answer

Stationary

Answer

Cannot be determined

Question 25

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

Accepted Answer

3x^2 - 4x + 5

Answer

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

Answer

3x^2 - 2x + 5

Answer

x^2 - 4x + 5

Question 26

Differentiate the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

2/(x^2 + 1)

Answer

x/(x^2 + 1)

Question 27

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

Accepted Answer

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

Answer

1

Answer

0

Answer

x

Question 28

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

Accepted Answer

e^x

Answer

e^x / x

Answer

xe^x

Answer

0

Question 29

Determine the derivative of f(x) = ln(x^2).

Accepted Answer

2/x

Answer

ln(x) / x

Answer

2x

Answer

1/x

Question 30

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

Accepted Answer

2cos(2x + 1)

Answer

cos(2x + 1)

Answer

-2cos(2x + 1)

Answer

-cos(2x + 1)

Question 31

Which of the following functions has a derivative equal to zero everywhere?

Accepted Answer

f(x) = c (where c is a constant)

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 32

Calculate the derivative of the function f(x) = ln(x^3 + 1).

Accepted Answer

f'(x) = (3x^2) / (x^3 + 1)

Answer

f'(x) = (x^3) / (x^3 + 1)

Answer

f'(x) = 3x^2

Answer

f'(x) = 3

Question 33

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

Accepted Answer

6x + 2

Answer

3x^2 + 4x - 2

Answer

9x^2 + 4x - 2

Answer

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

Question 34

Calculate the gradient of the normal to the curve y = x^3 at the point (1, 1).

Accepted Answer

-1/3

Answer

3

Answer

1/3

Question 35

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

Accepted Answer

-1

Answer

0

Answer

3

Answer

9

Question 36

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

Accepted Answer

3e^(3x - 2)

Answer

1/3e^(3x - 2)

Answer

-3e^(3x - 2)

Answer

e^(3x - 2)

Question 37

Differentiate the function f(x) = sin(2x + π/3).

Accepted Answer

2cos(2x + π/3)

Answer

cos(2x)

Answer

sin(2x)

Answer

2sin(2x)

Question 38

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

Accepted Answer

3e^(3x - 1)

Answer

e^3(3x - 1)

Answer

3xe^(3x - 1)

Question 39

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

Accepted Answer

12x^2 + 4

Answer

4x^3 + 4x

Answer

6x^2 + 2

Answer

12x^4 + 4x^2 - 3

Question 40

Determine the gradient of the tangent to the curve y = x^2 + 2x - 1 at the point (1, 2).

Accepted Answer

4

Answer

2

Answer

-2

Answer

6

Question 41

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

Accepted Answer

3e^(3x - 2)

Answer

e^(3x - 2)

Answer

-e^(3x - 2)

Answer

3x*e^(3x - 2)

Question 42

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

Accepted Answer

3x² + 4x - 5

Answer

3x² + 2x - 5

Answer

x² + 4x - 5

Answer

x³ + 4x - 5

Question 43

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

Accepted Answer

(2x² - 2x - 1)/(x - 1)²

Answer

x/(x - 1)²

Answer

(x² + 1)/(x - 1)

Answer

2x/(x - 1)²

Question 44

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

Accepted Answer

2x/(x² + 1)

Answer

2/(x² + 1)

Answer

x/(x² + 1)

Answer

1/(x² + 1)

Question 45

Calculate the derivative of the function f(x) = x^3 - 2x^2 + 5x - 7.

Accepted Answer

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

Answer

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

Answer

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

Question 46

Select the appropriate technique to differentiate an implicit function.

Accepted Answer

Implicit differentiation

Answer

Chain rule

Answer

Quotient rule

Answer

Product rule

Question 47

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

Accepted Answer

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

Answer

f'(x) = ln(x)

Answer

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

Answer

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