Master Derivatives in Calculus III: Interactive Quizzes and Practice

Question 1

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^3 + 4x - 5

Answer

x^3 + 4x - 5

Question 2

Which of the following is a commonly used differentiation rule for the product of two functions f(x) and g(x)?

Accepted Answer

Product Rule: (fg)'(x) = f'(x)g(x) + f(x)g'(x)

Answer

Chain Rule: (fg)'(x) = f'(g(x))g'(x)

Answer

Quotient Rule: (fg)'(x) = (f'(x)g(x) - f(x)g'(x)) / g(x)^2

Answer

Sum Rule: (fg)'(x) = f'(x) + g'(x)

Question 3

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

Accepted Answer

2x cos(x^2)

Answer

cos(x^2)

Answer

x cos(x^2)

Answer

2x^2 cos(x^2)

Question 4

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

Accepted Answer

Absolute maximum: 3, Absolute minimum: -1

Answer

Absolute maximum: 1, Absolute minimum: -3

Answer

Absolute maximum: 3, Absolute minimum: -3

Answer

Absolute maximum: 1, Absolute minimum: -1

Question 5

Calculate the derivative of the function y = sin(x²).

Accepted Answer

2x cos(x²)

Answer

cos(x²)

Answer

x² cos(x²)

Answer

None of the given options

Question 6

Determine the derivative of the function h(x) = e^(3x - 1).

Accepted Answer

3e^(3x - 1)

Answer

e^(3x)

Answer

3xe^(3x - 1)

Answer

None of the given options

Question 7

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

Accepted Answer

2x/(x² + 1)

Answer

1/(x² + 1)

Answer

(x² + 1)/x

Answer

None of the given options

Question 8

Find the second derivative of the function y = x⁴ - 3x² + 2x.

Accepted Answer

12x² - 6x

Answer

12x - 6

Answer

12x³ - 6x² + 2

Answer

None of the given options

Question 9

Find the maximum and minimum values of the function f(x) = x² - 4x + 3 on the interval [0, 3].

Accepted Answer

Minimum: -1, Maximum: 2

Answer

Minimum: 0, Maximum: 3

Answer

Minimum: 2, Maximum: 3

Answer

None of the given options

Question 10

Determine the derivative of the function f(x) = |x| using the limit definition of the derivative.

Accepted Answer

f'(x) = 1 if x > 0 and f'(x) = -1 if x < 0

Answer

f'(x) = 2x

Answer

f'(x) = |x - 1|

Answer

f'(x) = 0 for all x

Question 11

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

Accepted Answer

3x^2 - 4x

Answer

3x^2 + 4x

Answer

x^3 - 4x

Answer

x^3 + 4x

Question 12

Find the derivative of the function f(x) = sin(3x).

Accepted Answer

3cos(3x)

Answer

cos(3x)

Answer

-3cos(3x)

Answer

-sin(3x)

Question 13

Calculate the derivative of the function y = e^(2x+1).

Accepted Answer

2e^(2x+1)

Answer

2e^(2x)

Answer

e^(2x+1)

Answer

e^(2x)

Question 14

Calculate the derivative of the function y = sqrt(x^3 + 2).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

Calculate the derivative of the function y = arctan(2x).

Accepted Answer

2/(1 + 4x^2)

Answer

4/(1 + 4x^2)

Answer

1/(1 + 4x^2)

Answer

1/(2 + 4x^2)

Question 16

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

1/2e^(2x)

Answer

2xe^(2x)

Question 17

Evaluate the limit using l'Hôpital's rule: lim_(x -> 0) (e^x - 1) / x

Accepted Answer

1

Answer

0

Answer

e

Answer

(e^x - 1)

Question 18

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

Accepted Answer

[(2x - 1)(2x + 3) - (x^2 + 3x - 5)(2)] / (2x - 1)^2

Answer

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

Answer

(2x - 1) / (x^2 + 3x - 5)^2

Question 19

Which formula represents the chain rule for derivatives?

Accepted Answer

If g(x) = h(f(x)), then g'(x) = h'(f(x)) * f'(x)

Answer

If g(x) = h(f(x)), then g'(x) = h'(x) + f'(x)

Answer

If g(x) = h(f(x)), then g'(x) = h(f'(x))

Answer

If g(x) = h(f(x)), then g'(x) = f(h(x))

Question 20

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

Accepted Answer

(2x(x - 1) - (x² + 1))/(x - 1)²

Answer

((x² + 1) - x²)/(x - 1)²

Answer

2x

Answer

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

Question 21

If the derivative of f(x) is positive for all values of x, which characteristic describes f(x)?

Accepted Answer

Increasing

Answer

Decreasing

Answer

Constant

Answer

Concave up

Question 22

Find the maximum value of f(x) = -x² + 4x + 1 on the interval [0, 3].

Accepted Answer

5

Answer

3

Answer

0

Answer

1

Question 23

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

Accepted Answer

1/x

Answer

-1/x

Answer

x²

Answer

x

Question 24

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 2 + 5

Answer

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

Answer

3x - 4

Question 25

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

Accepted Answer

cos(x)

Answer

cos(x) - sin(x)

Answer

- cos(x)

Answer

sin(x) + cos(x)

Question 26

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

Accepted Answer

2e^(2x)

Answer

2x*e^(2x)

Answer

2e^(x^2)

Answer

e^x

Question 27

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

Accepted Answer

2x/(x^2 + 1)

Answer

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

Answer

(x^2 + 1)/2x

Question 28

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

Accepted Answer

1/(2√(x - 1))

Answer

2√(x - 1)

Answer

√(x - 1)/2

Answer

-1/(2√(x - 1))

Question 29

Find the equation of the tangent line to the curve y = x^3 - 2x^2 + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = 2x - 1

Answer

y = -x - 1

Answer

y = x + 1

Question 30

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

Accepted Answer

y' = cos(x)

Answer

y' = -sin(x)

Answer

y' = xcos(x)

Answer

y' = tan(x)

Question 31

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

Accepted Answer

f'(x) = e^x

Answer

f'(x) = 1/e^x

Answer

f'(x) = xe^x

Answer

f'(x) = -e^x

Question 32

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

Accepted Answer

y' = 1/x

Answer

y' = ln(x)

Answer

y' = x

Question 33

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

Accepted Answer

f'(x) = 2x * sin(x) + x² * cos(x)

Answer

f'(x) = x * sin(x) + x² * cos(x)

Answer

f'(x) = 2x * cos(x) - x² * sin(x)

Question 34

What is the derivative of y = (x² + 1)³?

Accepted Answer

y' = 3(x² + 1)² * 2x

Answer

y' = (x² + 1)²

Answer

y' = 3(x² + 1)³ / 2x

Answer

y' = 3x(x² + 1)²

Question 35

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

Accepted Answer

f'(x) = 1 (x > 0), f'(x) = -1 (x < 0), f'(0) does not exist

Answer

f'(x) = 1, f'(0) does not exist

Answer

f'(x) = 0

Question 36

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

Accepted Answer

y' = 1/(1 + x²)

Answer

y' = 1/x

Answer

y' = x/(1 + x²)

Answer

y' = -1/(1 + x²)

Question 37

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

Accepted Answer

f'(x) = 1/(2 * sqrt(x))

Answer

f'(x) = x * sqrt(x)

Answer

f'(x) = 2 * sqrt(x)

Answer

f'(x) = 1/sqrt(x)

Question 38

What is the derivative of y = ln(x² + 1)?

Accepted Answer

y' = 2x/(x² + 1)

Answer

y' = 1/(x² + 1)

Answer

y' = x/(x² + 1)

Answer

y' = 2x

Question 39

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

Accepted Answer

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

Answer

f'(x) = 3x^4 - 8x^3 + 5

Answer

f'(x) = 3x^2 - 4x - 7

Answer

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

Question 40

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

Accepted Answer

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

Answer

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

Answer

f'(x) = 2x * ln(x^2 + 1)

Answer

f'(x) = (2x) / (2x)

Question 41

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 42

Determine the equation of the tangent line to the curve y = x^2 - 3x + 2 at the point (1, 0).

Accepted Answer

y = -x + 1

Answer

y = -x - 1

Answer

y = x + 1

Question 43

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

Accepted Answer

f'(x) = e^x

Answer

f'(x) = -e^x

Answer

f'(x) = xe^x

Answer

f'(x) = 1/e^x

Question 44

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 45

Identify the absolute minimum of the function f(x) = x^3 - 6x^2 + 9x + 2 on the interval [0, 3].

Accepted Answer

Absolute minimum: f(1) = 4

Answer

Absolute minimum: f(3) = 8

Answer

Absolute minimum: f(2) = 6

Answer

Absolute minimum: f(0) = 2

Question 46

Let f(x) be a differentiable function. If f'(x) = 0 for all x, then f(x) is:

Accepted Answer

A constant function.

Answer

A quadratic function.

Answer

An exponential function.

Answer

A linear function.

Question 47

If the second derivative of a function *f(x)* is negative for all values of *x*, what can we conclude about the graph of *f(x)*?

Accepted Answer

The graph is concave down everywhere.

Answer

The graph has an inflection point at *x = 0*.

Answer

The graph is concave up everywhere.

Answer

The graph is always increasing.

Question 48

Using the chain rule, find the derivative of *h(x) = e^(2x - 1)*.

Accepted Answer

2e^(2x - 1)

Answer

e^(2x - 1)

Answer

1/2e^(2x - 1)

Answer

-2e^(2x - 1)

Question 49

What is the derivative of *f(x) = ln(x² + 1)*?

Accepted Answer

2x/(x² + 1)

Answer

x/(2x² + 1)

Answer

2/(x² + 1)

Answer

1/(x² + 1)

Question 50

Using the first derivative test, identify the local extrema of the function *g(x) = x³ + 3x² - 4*.

Accepted Answer

Local minimum at (-2, -4), local maximum at (-1, -3)

Answer

Local minimum at (-1, -3), local maximum at (-2, -4)

Answer

Local minimum at (-2, -4), local minimum at (-1, -3)

Answer

Local minimum at (1, -3), local maximum at (2, -4)

Question 51

Determine whether the function *f(x) = x³ - 6x² + 9x + 1* is increasing or decreasing on the interval (0, 2).

Accepted Answer

Increasing

Answer

Decreasing

Answer

Constant

Answer

Cannot be determined