Differential Calculus: Master Rates of Change in Mathematics

Question 1

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

Accepted Answer

3x² - 4x + 5

Answer

x³ - 4x² + 10x - 1

Answer

3x² + 4x + 5

Answer

x³ - 2x + 5

Question 2

Calculate the derivative of the function f(x) = (x^2 + 1) multiplied by e to the power of 3x.

Accepted Answer

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

Answer

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

Answer

3x^2 * e^(3x)

Answer

x^2 * e^(3x)

Question 3

Find the derivative of f(x) = sin(x^2) using the Chain Rule.

Accepted Answer

2x cos(x^2)

Answer

(2x) * (cos(x^2))

Answer

cos(x^2)

Answer

2x sin(x^2)

Question 4

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 5

Use derivatives to determine if the sequence 1, 4, 9, 16, 25, ... is increasing, decreasing, or neither.

Accepted Answer

Increasing

Answer

Decreasing

Answer

Neither

Question 6

To find the derivative of the function f(x) = e^x, which differentiation technique is used?

Accepted Answer

Chain rule

Answer

Power rule

Answer

Product rule

Answer

Quotient rule

Question 7

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2x * e^(2x)

Answer

x * e^(2x)

Question 8

Identify the critical point of f(x) = x² - 4x + 3.

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

x = 1

Question 9

What is the Chain Rule primarily used for?

Accepted Answer

Finding the derivative of composite functions

Answer

Finding the derivative of inverse functions

Answer

Finding the derivative of implicit functions

Answer

Finding the derivative of logarithmic functions

Question 10

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

Accepted Answer

3x² - 4x

Answer

4x³ - 4x

Answer

3x² + 4x

Answer

4x³ + 4x

Question 11

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

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Answer

-cos(2x)

Question 12

Sketch the curve of the function f(x) = x³ - 6x² + 9x.

Accepted Answer

A cubic curve with a local maximum and minimum

Answer

A parabola opening upward

Answer

A parabola opening downward

Answer

A cubic curve with an inflection point

Question 13

Find the slope of the tangent line to the curve y = x² + 2x - 1 at the point (1, 0).

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 14

Using the power rule, calculate the derivative of f(x) = x^2 + 3x - 5.

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

x + 3

Answer

2x - 5

Question 15

Employ the chain rule to determine the first derivative of f(x) = sin(x).

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sec(x)

Answer

1

Question 16

Determine the slope of the tangent line to the curve y = sin(x) at x = π/4.

Accepted Answer

√2/2

Answer

1

Answer

0

Answer

-1

Question 17

Which of the following is not a method for finding derivatives?

Accepted Answer

Integration

Answer

Power Rule

Answer

Chain Rule

Answer

Quotient Rule

Question 18

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

Accepted Answer

2x/(x² + 1)

Answer

1/(x² + 1)

Answer

x/(x² + 1)

Answer

(x² + 1)/x

Question 19

Which of the following is a characteristic of a decreasing function?

Accepted Answer

Negative derivative

Answer

Positive derivative

Answer

Zero derivative

Answer

Constant derivative

Question 20

Determine the equation of the tangent line to the curve y = e^x at the point (1, e).

Accepted Answer

y = e + e(x - 1)

Answer

y = e + x

Answer

y = e^x

Answer

y = e - e(x - 1)

Question 21

Which of the following is not a property of the derivative?

Accepted Answer

Symmetry

Answer

Linearity

Answer

Product Rule

Answer

Chain Rule

Question 22

Which of the following techniques is NOT used for finding derivatives?

Accepted Answer

Integration by parts

Answer

Power rule

Answer

Quotient rule

Answer

Chain rule

Question 23

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

xcos(x)

Answer

0

Question 24

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

Accepted Answer

e^x

Answer

xe^x

Answer

e

Answer

x

Question 25

Determine the absolute maximum value of the function f(x) = -x^2 + 4x - 5 on the interval [0, 3]:

Accepted Answer

2

Answer

-2

Answer

0

Answer

1

Question 26

Calculate the slope of the tangent line to the curve f(x) = x^3 - 2x^2 + 1 at x = 1?

Accepted Answer

3

Answer

1

Answer

-1

Answer

2

Question 27

Determine the critical points of the cubic function f(x) = x^3 - 6x^2 + 9x.

Accepted Answer

x = 0, x = 3

Answer

x = 0, x = 2

Answer

x = 1, x = 3

Answer

x = 2, x = 3

Question 28

Determine the rate of change of the volume of a sphere with respect to its radius.

Accepted Answer

4πr^2

Answer

3πr^2

Answer

4/3πr^2

Answer

3/4πr^2

Question 29

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

Accepted Answer

3x^2 - 4x

Answer

3x - 4

Answer

x^3 - 2

Answer

x^3 - 4x^2

Question 30

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

Accepted Answer

√2/2

Answer

1

Answer

-1

Question 31

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

Accepted Answer

2e^(2x)

Answer

e^(2)

Answer

e^(2x)/2

Answer

2x * e^(2x)

Question 32

Determine the local maximum of the function f(x) = -x^2 + 4x - 3.

Accepted Answer

(2, 1)

Answer

(0, -3)

Answer

(-2, 1)

Answer

(4, -3)

Question 33

Which technique is used to find the derivative of an implicit function?

Accepted Answer

Implicit differentiation

Answer

Quotient rule

Answer

Chain rule

Answer

Product rule

Question 34

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 35

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

Accepted Answer

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

Answer

f'(x) = sin(2x)

Answer

f'(x) = cos(2x)

Answer

f'(x) = -2sin(2x)

Question 36

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

Accepted Answer

f''(x) = e^x

Answer

f''(x) = 2e^x

Answer

f''(x) = x

Answer

f''(x) = 0

Question 37

What is the derivative of the inverse function of *f(x) = x^3 + 1*?

Accepted Answer

(f^-1)'(x) = 1/(3√(x^2 + 1))

Answer

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

Answer

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

Answer

(f^-1)'(x) = 3x^2

Question 38

If *f(x)* is differentiable and *f'(x) = 0* for all *x*, what kind of function is *f(x)*?

Accepted Answer

a constant function

Answer

a quadratic function

Answer

an exponential function

Answer

a linear function

Question 39

What are the critical points of the function *h(x) = x^3 - 6x^2 + 9x - 2*?

Accepted Answer

x = 1, x = 3

Answer

x = -1, x = 2

Answer

x = 0, x = 2

Answer

x = 0, x = 1

Question 40

What is the derivative of the inverse trigonometric function *f(x) = sin^-1(x)*?

Accepted Answer

f'(x) = 1/sqrt(1 - x^2)

Answer

f'(x) = -1/sqrt(1 - x^2)

Answer

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

Question 41

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x^2 - 5

Answer

3x^2 - 5x + 1

Answer

x^3 + 2x^2 + 1

Question 42

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

Accepted Answer

√2/2

Answer

1

Answer

0

Answer

-1

Question 43

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

Accepted Answer

x = 0, ±√2

Answer

x = -2, 2

Answer

x = -√2, 0, √2

Answer

x = 0, 2

Question 44

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

Accepted Answer

2xe^(x^2)

Answer

e^(x^2)

Answer

x^2e^(x^2)

Question 45

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

Accepted Answer

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

Answer

x^3/(x^3 + 1)

Answer

3x/(x^3 + 1)

Answer

1/(x^3 + 1)

Question 46

Which of the following is the derivative of f(x) = x³?

Accepted Answer

3x²

Answer

x²/3

Answer

x³

Answer

1/3x²

Question 47

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

Accepted Answer

cos(x) + e^x

Answer

-sin(x) + e^x

Answer

sin(x) - e^x

Answer

cos(x) - e^x

Question 48

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

Accepted Answer

x/|x|

Answer

1/|x|

Answer

-x/|x|

Answer

0

Question 49

If f(x) is differentiable and increasing on the interval (a, b), then what can you conclude?

Accepted Answer

f'(x) > 0

Answer

f'(x) = 0

Answer

f''(x) > 0

Answer

f'(x) < 0

Question 50

Find the critical points of the function f(x) = x³ - 3x² + 2.

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = -2

Answer

x = -1, x = 2

Answer

x = 1, x = 2