Master Derivatives in Calculus: Enhance Your Mathematical Skills

Question 1

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

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 2x - 5

Answer

x² - 2x - 5

Question 2

Find the derivative of f(x) = (x² + 1)(x - 2). (Product rule)

Accepted Answer

2x(x - 2) + (x² + 1)

Answer

(x² + 1)(x - 3)

Answer

2x(x - 3) + (x² + 1)

Answer

(x² - 1)(x - 2)

Question 3

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

Accepted Answer

0

Answer

c

Answer

-c

Answer

x

Question 4

Find the derivative of the function f(x) = ln(x² + 1). (Chain rule, logarithmic function)

Accepted Answer

2x/(x² + 1)

Answer

x/(x² + 1)

Answer

1/(x² + 1)

Answer

2/(x² + 1)

Question 5

To find the derivative of f(x) = x^3 using the power rule, what mathematical operation should be performed?

Accepted Answer

Multiply x^3 by 3

Answer

Divide x^3 by 3

Answer

Add 3 to x^3

Answer

Subtract 3 from x^3

Question 6

When applying the product rule to find the derivative of f(x) = (x^2 + 1)(x - 3), what are the two factors being multiplied together?

Accepted Answer

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

Answer

x^2 and x

Answer

x^2 and 1

Answer

x and 1

Question 7

Which of the following expressions correctly represents the derivative of f(x) = e^(2x)?

Accepted Answer

2 * e^(2x)

Answer

e^(2x) / 2

Answer

e^(x)

Answer

2 * e^(x)

Question 8

When finding the derivative of f(x) = ln(x^3) using the chain rule, what is the derivative of the outer function ln(u)?

Accepted Answer

1 / u

Answer

u

Answer

u^2

Answer

1 / u^2

Question 9

To find the second derivative of f(x) = x^4, what operation should be performed twice successively?

Accepted Answer

Differentiation

Answer

Integration

Answer

Substitution

Answer

Simplification

Question 10

To find the equation of the tangent line to the function f(x) = x^2 - 2x at the point (1, -1), what information is needed?

Accepted Answer

The slope of the function at the given point

Answer

The y-intercept of the function

Answer

The coordinates of another point on the tangent line

Answer

The value of the function at the given point

Question 11

To find the absolute extrema of the function f(x) = x^2 - 4x + 3 on the interval [0, 3], what should be evaluated?

Accepted Answer

The function's values at the endpoints and critical points

Answer

The function's derivatives at the endpoints and critical points

Answer

The function's integrals at the endpoints and critical points

Answer

The function's second derivatives at the endpoints and critical points

Question 12

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

Accepted Answer

2x cos(x^2)

Answer

sin(x^2)

Answer

cos(x^2)

Answer

2x sin(x^2)

Question 13

Which of the following is NOT a technique for finding derivatives?

Accepted Answer

Integration by parts

Answer

Power rule

Answer

Product rule

Answer

Chain rule

Question 14

Which of the following is the derivative of 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 15

Which of the following statements is true about the derivative of a constant function?

Accepted Answer

It is zero.

Answer

It is equal to the constant.

Answer

It is undefined.

Answer

It is equal to x.

Question 16

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

Accepted Answer

1/x

Answer

1

Answer

-1/x^2

Answer

x^2

Question 17

Which technique is used to find the derivative of f(x) = (x^2 - 1)/(x + 1)?

Accepted Answer

Quotient Rule

Answer

Power Rule

Answer

Product Rule

Answer

Chain Rule

Question 18

Determine the second derivative of f(x) = cos(x):

Accepted Answer

-cos(x)

Answer

-sin(x)

Answer

sin(x)

Answer

cos(x)

Question 19

What is the derivative of the function f(x) = x³ + 2x² - 5?

Accepted Answer

3x² + 4x

Answer

x² + 4x - 5

Answer

3x² + 2

Answer

x³ + 4x

Question 20

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

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(x)

Answer

2sin(x)

Question 21

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

Accepted Answer

2x / (x² + 1)

Answer

1 / (x² + 1)

Answer

ln(x)

Answer

2ln(x)

Question 22

Given a function f(x) with continuous second derivatives, if f''(x) is positive for all x in an interval (a, b), what can you conclude about the concavity of f(x) in (a, b)?

Accepted Answer

f(x) is concave up in (a, b).

Answer

f(x) is concave down in (a, b).

Answer

The concavity of f(x) in (a, b) cannot be determined from the given information.

Question 23

Use the product rule to find the derivative of the function g(x) = (x² + 1)(x - 2).

Accepted Answer

g'(x) = 2x(x - 2) + (x² + 1)

Answer

g'(x) = x(x² + 1) - 2(x - 2)

Answer

g'(x) = x(x - 2) + (x² + 1)

Answer

g'(x) = 2x(x² + 1) - (x - 2)

Question 24

Employ the chain rule to find the derivative of the function h(x) = sin(2x + 1).

Accepted Answer

h'(x) = 2cos(2x + 1)

Answer

h'(x) = cos(2x + 1)

Answer

h'(x) = sin(2x + 1) * 2

Answer

h'(x) = cos(2x + 1) * 2

Question 25

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

Accepted Answer

f'(x) = e^x

Answer

f'(x) = x * e^x

Answer

f'(x) = 1 / e^x

Answer

f'(x) = 0

Question 26

Which of the following is a necessary condition for a function to be differentiable at a point?

Accepted Answer

The function must be continuous at the point.

Answer

The function must have a non-zero derivative at the point.

Answer

The function must have a finite derivative at the point.

Answer

The function must have an inflection point at the point.

Question 27

Which of the following functions is not differentiable at (x = 0)?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 28

What is the relationship between the derivative of a function and its graph?

Accepted Answer

The derivative of a function gives the slope of the tangent line to the graph at any point.

Answer

The derivative of a function gives the inflection points of the graph.

Answer

The derivative of a function gives the maximum and minimum points of the graph.

Answer

The derivative of a function gives the area under the graph of the function.

Question 29

To differentiate a function that is the product of two differentiable functions, which derivative technique should be employed?

Accepted Answer

Product rule

Answer

Power rule

Answer

Chain rule

Answer

Quotient rule

Question 30

What does a negative sign in the derivative of a function indicate about the function's behavior?

Accepted Answer

The function is decreasing.

Answer

The function is increasing.

Answer

The function is constant.

Question 31

Identify the function that is not differentiable at x = 0:

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 32

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

Accepted Answer

3x^2 + 4x

Answer

x^2 + 2x

Answer

3x^3 - 4x

Answer

x^3 - 2x^2 + 5

Question 33

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

Accepted Answer

2x^2 - 5x - 3

Answer

x^2 - 2x - 3

Answer

2x^3 - 5x^2 - 3

Answer

x^3 - 2x^2 - 3

Question 34

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

Accepted Answer

2cos(2x + 1)

Answer

-2cos(2x + 1)

Answer

sin(2x - 1)

Answer

cos(2x + 1)

Question 35

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

Accepted Answer

e^x

Answer

e^(x^2)

Answer

x*e^x

Answer

(1/e)*x

Question 36

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

Accepted Answer

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

Answer

x^2

Answer

(1/x)

Answer

2ln(x)

Question 37

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

Accepted Answer

1 / (1 + x^2)

Answer

x / (1 + x^2)

Answer

1

Answer

-arctan(x)

Question 38

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

Accepted Answer

6x + 2

Answer

6x² + 2x

Answer

9x + 2

Answer

3x + 2

Question 39

The derivative of a function at a specific point represents:

Accepted Answer

The instantaneous rate of change of the function at that point.

Answer

The maximum value of the function.

Answer

The average rate of change of the function over a small interval.

Question 40

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

Accepted Answer

e^x

Answer

x*e^(x-1)

Answer

1/e^x

Answer

e^(x-1)

Question 41

The derivative of the function *f(x) = ln(x)* is:

Accepted Answer

1/x

Answer

ln(x)

Answer

x

Answer

x*ln(x)

Question 42

A particle's position is described by the function *s(t) = 3t² - 5t + 2*, where *s* represents the position in meters and *t* represents time in seconds. What is the particle's instantaneous velocity at *t = 2* seconds?

Accepted Answer

7 m/s

Answer

11 m/s

Answer

5 m/s

Question 43

What is the derivative of the function *f(x) = x³ - 4x² + 7x - 2*?

Accepted Answer

f'(x) = 3x² - 8x + 7

Answer

f'(x) = x² - 8x + 7

Answer

f'(x) = 3x² - 4x + 7

Answer

f'(x) = 3x² - 8x + 5

Question 44

Calculate the derivative of the function *g(x) = e^2x*.

Accepted Answer

g'(x) = 2e^2x

Answer

g'(x) = e^2x

Answer

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

Answer

g'(x) = 2xe^2x-1

Question 45

What is the derivative of *y = ln(5x)*?

Accepted Answer

y' = 1/x

Answer

y' = ln(5)/x

Answer

y' = 5/x

Answer

y' = 1/(5x)

Question 46

If *h(x) = sin(3x + 2)*, what is *h'(x)*?

Accepted Answer

h'(x) = 3cos(3x + 2)

Answer

h'(x) = cos(3x + 2)

Answer

h'(x) = -3cos(3x + 2)

Answer

h'(x) = 3sin(3x + 2)

Question 47

What is the slope of the tangent line to the curve *y = x² + 3x - 1* at the point *(1, 3)*?

Accepted Answer

5

Answer

2

Answer

3

Question 48

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

Accepted Answer

f'(x) = 6x² - 6x + 2

Answer

f'(x) = 4x - 6

Answer

f'(x) = 6x² - 6x - 3

Answer

f'(x) = 2x³ - 3x² + 2x - 3

Question 49

What is the derivative of *g(x) = (x + 1)/(x - 1)*?

Accepted Answer

g'(x) = -2/(x - 1)²

Answer

g'(x) = 1/(x - 1)²

Answer

g'(x) = -2/(x + 1)²

Answer

g'(x) = 2/(x - 1)²

Question 50

A spherical balloon is inflated so that its radius increases at a rate of 2 cm/s.  At what rate is the volume of the balloon increasing when the radius is 5 cm?

Accepted Answer

200π cm³/s

Answer

100π cm³/s

Answer

50π cm³/s

Question 51

Determine the critical points of the function *f(x) = x³ - 6x² + 9x*.

Accepted Answer

x = 1, x = 3

Answer

x = 0, x = 2

Answer

x = 0, x = 3