Master Derivatives: Enhance Your Calculus Skills

Question 1

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

Accepted Answer

3x²

Answer

x²

Answer

3x

Answer

x⁴

Question 2

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

Accepted Answer

2xe^(x²)

Answer

x²e^(x²)

Answer

e^(x²)

Answer

e^(2x)

Question 3

Which of the following functions has a negative first derivative on the interval (-∞, 0)?

Accepted Answer

g(x) = -x²

Answer

f(x) = x³

Answer

h(x) = sin(x)

Answer

k(x) = cos(x)

Question 4

Which of the following is a necessary condition for a function to have an inflection point?

Accepted Answer

Second derivative changes sign.

Answer

First derivative is zero.

Answer

First derivative is non-zero.

Answer

Second derivative is zero.

Question 5

Determine the derivative of the function g(x) = x^2 + 2x - 5.

Accepted Answer

f'(x) = 2x + 2

Answer

f'(x) = x^2 + 2

Answer

f'(x) = 2x - 5

Answer

f'(x) = x^2 - 10x + 25

Question 6

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

Accepted Answer

h'(x) = cos(x)

Answer

h'(x) = sin(x)

Answer

h'(x) = -cos(x)

Answer

h'(x) = -sin(x)

Question 7

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

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x - 5

Answer

3x² - 5x + 1

Question 8

Find the derivative of h(x) = x * e^x using the Product Rule.

Accepted Answer

e^x + x * e^x

Answer

e^x

Answer

x * e^x

Answer

e^x + x

Question 9

Find the derivative of the function f(x) = x^3 using the power rule.

Accepted Answer

f'(x) = 3x^2

Answer

f'(x) = 2x

Answer

f'(x) = 3x

Answer

f'(x) = x^2

Question 10

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

Accepted Answer

f'(x) = cos(x)

Answer

f'(x) = sin(x)

Answer

f'(x) = -cos(x)

Answer

f'(x) = -sin(x)

Question 11

Find the derivative of the function f(x) = ln(x) using the natural logarithmic rule.

Accepted Answer

f'(x) = 1/x

Answer

f'(x) = ln(x)

Answer

f'(x) = x

Answer

f'(x) = -1/x

Question 12

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

Accepted Answer

f'(x) = e^x

Answer

f'(x) = x

Answer

f'(x) = -e^x

Answer

f'(x) = 1/e^x

Question 13

Find the derivative of the function f(x) = x^2 + 3x - 2 using the sum/difference rule.

Accepted Answer

f'(x) = 2x + 3

Answer

f'(x) = x^2 + 3

Answer

f'(x) = 2x

Answer

f'(x) = 3x - 2

Question 14

Find the derivative of the function f(x) = sqrt(x) using the chain rule.

Accepted Answer

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

Answer

f'(x) = 2sqrt(x)

Answer

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

Answer

f'(x) = sqrt(x)

Question 15

Find the derivative of the function f(x) = 1/x using the quotient rule.

Accepted Answer

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

Answer

f'(x) = 1/x^2

Answer

f'(x) = -x

Answer

f'(x) = x

Question 16

The derivative of any constant function is always equal to:

Accepted Answer

0

Answer

1

Answer

The constant itself

Answer

Infinity

Question 17

Which of the following is NOT a common method used to calculate the derivative of a function?

Accepted Answer

Integration by parts

Answer

Power Rule

Answer

Chain Rule

Answer

Product Rule

Question 18

Find the derivative of the function f(x) = x³ + 2x² - 5 using the Power Rule:

Accepted Answer

3x² + 4x

Answer

3x + 2

Answer

x² + 4x

Answer

3x³ + 4x²

Question 19

The derivative of a polynomial function is always a polynomial function of:

Accepted Answer

One degree higher

Answer

One degree lower

Answer

The same degree

Answer

Two degrees higher

Question 20

Find the derivative of the function f(x) = sin(x) + cos(x) using the Chain Rule:

Accepted Answer

cos(x) + sin(x)

Answer

cos(x) - sin(x)

Answer

sin(x) + cos(x)

Answer

-sin(x) - cos(x)

Question 21

Calculate the derivative of the function g(x) = sin(x) using the chain rule.

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Answer

cos(x)/sin(x)

Question 22

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2

Answer

e^x

Question 23

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

Accepted Answer

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

Answer

x if x > 0, 0 if x = 0, -x if x < 0

Answer

x

Answer

0

Question 24

Determine the derivative of the function g(x) = sqrt(x).

Accepted Answer

1/(2sqrt(x))

Answer

2sqrt(x)

Answer

1/sqrt(x)

Answer

x^2

Question 25

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

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

0

Question 26

Calculate the rate of change of the function f(x) = x^2 + 2x - 3 at x = 5.

Accepted Answer

15

Answer

12

Answer

9

Answer

6

Question 27

Which of the following describes the relationship between a function and its derivative?

Accepted Answer

The derivative is the slope of the function at a given point.

Answer

The derivative is the maximum value of the function.

Answer

The derivative is the area under the curve of the function.

Answer

The derivative is the inverse of the function.

Question 28

Which theorem is the foundation of calculus, linking derivatives and integrals, allowing us to find the area under curves?

Accepted Answer

Fundamental Theorem of Calculus

Answer

Chain Rule

Answer

Product Rule

Answer

Quotient Rule

Question 29

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

Accepted Answer

3cos(3x - 2)

Answer

sin(3x - 2)/3

Answer

-3sin(3x - 2)

Answer

cos(3x - 2)

Question 30

Determine the derivative of the function f(x) = |x - 1|.

Accepted Answer

1 if x > 1, -1 if x < 1

Answer

0

Answer

-1 if x > 1, 1 if x < 1

Answer

|x + 1|

Question 31

Determine the interval(s) where the function f(x) = x⁴ - 2x² + 1 is increasing.

Accepted Answer

(-∞, -1) U (1, ∞)

Answer

(-∞, 1)

Answer

(-1, 1)

Question 32

Determine 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

1/(x - 1)

Answer

(x² + 1)/(x - 1)

Question 33

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

Accepted Answer

f'(x) = 3x² + 4x - 5

Answer

f'(x) = 3x³ + 2x² - 5

Answer

f'(x) = x² + 4x - 5

Answer

f'(x) = 3x² + 2x + 5

Question 34

Calculate the derivative of g(x) = sin(x).

Accepted Answer

g'(x) = cos(x)

Answer

g'(x) = xcos(x)

Answer

g'(x) = -sin(x)

Answer

g'(x) = tan(x)

Question 35

Find the slope of the tangent line to the curve y = x² - 3x + 2 at the point (1, 0).

Accepted Answer

-1

Answer

-2

Answer

1

Answer

2

Question 36

Identify the critical points of f(x) = (x - 1)²(x + 2).

Accepted Answer

x = 1, x = -2

Answer

x = 0, x = 1

Answer

x = -2

Answer

x = 1

Question 37

Which function's derivative equals 2x?

Accepted Answer

f(x) = x^2

Answer

h(x) = sin(x)

Answer

g(x) = e^x

Question 38

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

Accepted Answer

y' = 2e^(2x + 3)

Answer

y' = e^(2x + 3)

Answer

y' = 2x.e^(2x + 3)

Answer

y' = 3e^(2x + 3)

Question 39

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 40

Which function has a negative derivative for all real numbers?

Accepted Answer

f(x) = -x^2

Answer

j(x) = sin(x)

Answer

g(x) = x^3

Answer

h(x) = e^x

Question 41

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

Accepted Answer

3x^2 + 4x

Answer

x^2 + 2x - 1

Answer

3x - 2

Answer

x^3 - 2x

Question 42

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

Accepted Answer

2e^(2x)

Answer

e^(2x) / 2

Answer

1 / e^(2x)

Answer

0

Question 43

Which of the following is a geometric interpretation of the derivative of a function at a given point?

Accepted Answer

The slope of the tangent line to the graph of the function at that point

Answer

The area under the curve of the function between two points

Answer

The average rate of change of the function over an interval

Question 44

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

Accepted Answer

2 / x

Answer

1 / x^2

Answer

-1 / x^2

Answer

0

Question 45

Which of the following techniques can be used to find the derivative of an inverse function?

Accepted Answer

Inverse Function Theorem

Answer

Chain Rule only

Answer

Product Rule

Question 46

Find the derivative of the function f(x) = |x - 2|.

Accepted Answer

1 for x > 2, -1 for x < 2

Answer

2 for x > 2, -2 for x < 2

Answer

|x - 2| for all x

Answer

0 for all x

Question 47

Calculate the derivative of y = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

2sin(2x)

Answer

-2cos(2x)

Question 48

What is the derivative of the function f(x) = e^(3x)?

Accepted Answer

3e^(3x)

Answer

3xe^(3x)

Answer

-3e^(3x)

Answer

e^(3x)

Question 49

If a function f(x) is differentiable and its derivative f'(x) = 0 for all values of x, what type of function is f(x)?

Accepted Answer

Constant function

Answer

Exponential function

Answer

Quadratic function

Answer

Linear function with a non-zero slope

Question 50

What is the second derivative of the function f(x) = x^4 - 3x^2 + 5?

Accepted Answer

12x^2 - 6

Answer

x^4 - 3x^2 + 5

Answer

12x - 6

Answer

4x^3 - 6x