Calculus 1 Quiz: Basic Derivative Rules

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Basic Derivative RulesQuestion 1 of 20

Let f(x) and g(x) be differentiable functions with h'(x) = 5f'(x) - 2g'(x). If h(0) = 1, f(0) = 2, and g(0) = 3, which of the following could be an expression for h(x)?

5f(x) - 2g(x)

5f(x) - 2g(x) + 3

5f(x) - 2g(x) - 4

5f(x) - 2g(x) - 3

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Calculus 1 Quiz

Calculus 1 Quiz: Basic Derivative Rules

Practice Basic Derivative Rules in Calculus 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Basic Derivative Rules, giving you a quick way to practice the rules, question types, and explanations that matter most for Calculus 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let f(x) and g(x) be differentiable functions with h'(x) = 5f'(x) - 2g'(x). If h(0) = 1, f(0) = 2, and g(0) = 3, which of the following could be an expression for h(x)?

5f(x) - 2g(x)
5f(x) - 2g(x) + 3
5f(x) - 2g(x) - 4
5f(x) - 2g(x) - 3 (correct answer)

Explanation: This problem tests your understanding of the relationship between derivatives and antiderivatives, specifically how differentiation rules work in reverse. Given that

h'(x) = 5f'(x) - 2g'(x)

, you need to find which expression for

h(x)

has this derivative. Using the linearity of differentiation, if

h(x) = 5f(x) - 2g(x) + C

(where

C

is a constant), then

h'(x) = 5f'(x) - 2g'(x)

since the derivative of a constant is zero. This confirms the general form must be

5f(x) - 2g(x) + C

. To find the specific value of

C

, use the initial conditions. You know

h(0) = 1,f(0) = 2

, and

g(0) = 3

. Substituting into

h(x) = 5f(x) - 2g(x) + C

1 = 5(2) - 2(3) + C

1 = 10 - 6 + C

1 = 4 + C

C = -3

Therefore,

h(x) = 5f(x) - 2g(x) - 3

, which is answer choice D. Choice A lacks the constant term needed to satisfy the initial condition. Choice B gives

h(0) = 5(2) - 2(3) + 3 = 7 \neq 1

. Choice C gives

h(0) = 5(2) - 2(3) - 4 = 0 \neq 1

. Study tip: When working backwards from a derivative to find the original function, remember that antiderivatives always include a constant of integration. Use initial conditions to determine this constant's specific value.

Question 2

Let D be the differentiation operator, such that D(f) = f'. Which of the following statements best describes a property of this operator for any differentiable functions f and g and any constants a and b?

D(a·f + b·g) = a·D(f) + b·D(g) (correct answer)
D(f + g) = D(f)·D(g)
D(a·f) = a + D(f)
D(f·g) = D(f)·g + f·D(g)

Explanation: This question asks to identify the linearity property of differentiation. The Sum Rule states D(f+g) = D(f) + D(g), and the Constant Multiple Rule states D(c·f) = c·D(f). Combining these, we get D(a·f + b·g) = D(a·f) + D(b·g) = a·D(f) + b·D(g). This property is known as linearity. Option D is the product rule, which is a different property.

Question 3

A function is defined by the polynomial $p(x) = ax^3 + bx^2 + cx + d$ . The function satisfies the conditions $p(0) = 5$ , $p'(0) = -3$ , and $p''(0) = 8$ . What is the value of the coefficient $b$ ?

-3
2
4 (correct answer)
8

Explanation: We find the first and second derivatives of the general polynomial

p(x)

.\n

p'(x) = 3ax^2 + 2bx + c

p''(x) = 6ax + 2b

\nNow we use the given conditions. The condition

p''(0)=8

allows us to find

b

. \n

p''(0) = 6a(0) + 2b = 8

2b = 8

b = 4

\nThe other conditions determine the other coefficients:

p(0) = d = 5

and

p'(0) = c = -3

. The question only asks for the value of

b

Question 4

Let h(t) = C₁cos(t) + C₂sin(t) for constants C₁ and C₂. Which of the following is equal to the second derivative, h''(t)? Assume the derivative of sin(t) is cos(t) and the derivative of cos(t) is -sin(t).

h(t)
-h(t) (correct answer)
h'(t)
0

Explanation: First, we find the first derivative, h'(t). Using the derivative rules, h'(t) = C₁(-sin(t)) + C₂cos(t) = -C₁sin(t) + C₂cos(t). Next, we differentiate h'(t) to find the second derivative, h''(t). h''(t) = -C₁cos(t) + C₂(-sin(t)) = -C₁cos(t) - C₂sin(t). We can factor out -1 from this expression: h''(t) = -(C₁cos(t) + C₂sin(t)). Since the expression in the parentheses is the original function h(t), we have h''(t) = -h(t).

Question 5

Let $f(x)$ be a differentiable function and let $A$ and $B$ be constants. A function $h(x)$ is defined as $h(x) = A \cdot f(x) + B$ . Given that $h(2) = 10$ , $h'(2) = 12$ , $f(2) = 1$ , and $f'(2) = 3$ , what is the value of the constant $A$ ?

3
4 (correct answer)
6
9

Explanation: The question asks for the value of the constant

A

. We need to use the information about the derivatives. First, find the derivative of

h(x)

using the constant multiple and constant rules:

h'(x) = A \cdot f'(x) + 0 = A \cdot f'(x)

. Now, substitute the given values at

x=2

h'(2) = A \cdot f'(2)

. We are given

h'(2)=12

and

f'(2)=3

. So, the equation becomes

12 = A \cdot 3

. Solving for

A

, we get

A = 12 / 3 = 4

. The information

h(2)=10

and

f(2)=1

is extra information not needed to find

A

Question 6

Let $f(x)$ and $g(x)$ be differentiable functions. At $x=2$ , the following values are known: $f(2)=3$ , $f'(2)=-1$ , $g(2)=5$ , and $g'(2)=2$ . If $h(x) = 4f(x) - g(x)$ , what is the equation of the tangent line to the graph of $h(x)$ at $x=2$ ?

$y = -6x + 15$
$y = -2x + 11$
$y = -6x + 19$ (correct answer)
$y = 7x - 7$

Explanation: The equation of the tangent line to

h(x)

x=2

is given by

y - h(2) = h'(2)(x - 2)

. We need to find both

h(2)

and

h'(2)

. First, find the value of the function at

x=2

h(2) = 4f(2) - g(2) = 4(3) - 5 = 12 - 5 = 7

. So the point of tangency is

(2, 7)

. Next, find the derivative of

h(x)

h'(x) = 4f'(x) - g'(x)

. Evaluate the derivative at

x=2

to find the slope of the tangent line:

h'(2) = 4f'(2) - g'(2) = 4(-1) - 2 = -4 - 2 = -6

. Now substitute the point

(2, 7)

and the slope

m=-6

into the point-slope form:

y - 7 = -6(x - 2)

. Distributing the slope gives

y - 7 = -6x + 12

. Adding 7 to both sides gives the final equation:

y = -6x + 19

Question 7

Let $f(x)$ and $g(x)$ be differentiable functions for all real numbers. If $f'(x) = g'(x)$ for all $x$ , and $h(x) = f(x) - g(x)$ , which of the following statements must be true about the graph of $y=h(x)$ ?

The graph is the line $y=0$ .
The graph is a vertical line.
The graph is a horizontal line. (correct answer)
The graph is a line with a slope of 1.

Explanation: We are given the function

h(x) = f(x) - g(x)

. To understand the graph of

h(x)

, we can analyze its derivative,

h'(x)

. Using the difference rule for derivatives, we have

h'(x) = f'(x) - g'(x)

. The problem states that

f'(x) = g'(x)

for all

x

. Substituting this into the equation for

h'(x)

gives:

h'(x) = g'(x) - g'(x) = 0

. A function whose derivative is zero for all

x

in an interval must be a constant function on that interval. So,

h(x) = C

for some constant

C

. The graph of a constant function

y=C

is a horizontal line. The constant

C

is not necessarily zero.

Question 8

Let $f, g,$ and $k$ be differentiable functions. Define two new functions, $h(x) = 2f(x) - 3g(x)$ and $p(x) = 5g(x) + k(x)$ . If $f'(1) = 4$ , $h'(1) = -1$ , and $k'(1) = 7$ , what is the value of $p'(1)$ ?

-8
$56/3$
27
22 (correct answer)

Explanation: This problem tests your understanding of differentiation rules, specifically how to find derivatives of linear combinations of functions and how to work with given derivative information. When you have

h(x) = 2f(x) - 3g(x)

, the derivative is

h'(x) = 2f'(x) - 3g'(x)

by the constant multiple and difference rules. At

x = 1

h'(1) = 2f'(1) - 3g'(1) = 2(4) - 3g'(1) = 8 - 3g'(1)

. Since

h'(1) = -1

, you get

-1 = 8 - 3g'(1)

, which gives

g'(1) = 3

. For

p(x) = 5g(x) + k(x)

, you have

p'(x) = 5g'(x) + k'(x)

. Therefore:

p'(1) = 5g'(1) + k'(1) = 5(3) + 7 = 22

, which is answer choice D. Let's examine the wrong answers: A) -8 likely comes from incorrectly calculating

g'(1)

or making sign errors in the algebra. B) 56/3 might result from division errors when solving for

g'(1)

or incorrectly setting up the derivative equations. C) 27 could come from adding

k'(1) = 7

to an incorrect value like 20, possibly from mishandling the coefficients. Study tip: When working with derivative problems involving multiple functions, always write out the derivative rules explicitly first, then substitute known values systematically. Work step-by-step to find any missing derivatives before calculating your final answer—rushing to plug in values often leads to algebraic mistakes.

Question 9

Let $f(x)$ and $g(x)$ be differentiable functions. Define the sum function $S(x) = f(x) + g(x)$ and the difference function $D(x) = f(x) - g(x)$ . If $S'(a) = 10$ and $D'(a) = 2$ for some constant $a$ , what is the value of $f'(a)$ ?

4
6 (correct answer)
8
12

Explanation: First, we find the derivatives of

S(x)

and

D(x)

using the sum and difference rules:

S'(x) = f'(x) + g'(x)

D'(x) = f'(x) - g'(x)

We are given the values of these derivatives at

x=a

. This gives us a system of two linear equations with two unknowns,

f'(a)

and

g'(a)

$f'(a) + g'(a) = 10$
$f'(a) - g'(a) = 2$

We want to solve for

f'(a)

. We can do this by adding the two equations together. The

g'(a)

terms will cancel out:

(f'(a) + g'(a)) + (f'(a) - g'(a)) = 10 + 2

2f'(a) = 12

Dividing by 2, we get:

f'(a) = 6

Question 10

Let $f(x)$ and $g(x)$ be differentiable functions. Two new functions are defined as $h_1(x) = 2f(x) + 3g(x)$ and $h_2(x) = 3f(x) + 2g(x)$ . Suppose that for some constant $c$ , we have $f'(c) > g'(c) > 0$ . Which of the following statements must be true?

$h_2'(c) > h_1'(c)$ (correct answer)
$h_1'(c) > h_2'(c)$
$h_1'(c) = h_2'(c)$
The relationship cannot be determined from the information given.

Explanation: When you see functions combined linearly like this, you're dealing with differentiation rules for linear combinations. The key insight is recognizing how coefficients affect the derivatives when you have competing rates of change. First, let's find the derivatives using the linearity of differentiation. For

h_1(x) = 2f(x) + 3g(x)

, we get

h_1'(x) = 2f'(x) + 3g'(x)

. For

h_2(x) = 3f(x) + 2g(x)

, we get

h_2'(x) = 3f'(x) + 2g'(x)

. At point

c

, this gives us

h_1'(c) = 2f'(c) + 3g'(c)

and

h_2'(c) = 3f'(c) + 2g'(c)

. Since

f'(c) > g'(c)

, we can compare these expressions by looking at their difference:

h_2'(c) - h_1'(c) = [3f'(c) + 2g'(c)] - [2f'(c) + 3g'(c)] = f'(c) - g'(c)

. Because

f'(c) > g'(c)

, we have

f'(c) - g'(c) > 0

, which means

h_2'(c) > h_1'(c)

. Choice A is correct because

h_2

gives more weight (coefficient 3) to the faster-growing function

f

. Choice B reverses this relationship incorrectly. Choice C would only be true if

f'(c) = g'(c)

, which contradicts our given condition. Choice D is wrong because we have sufficient information to determine the relationship definitively. Study tip: When comparing linear combinations of functions, look at which combination gives more weight to the terms with larger derivatives—that combination will have the larger overall derivative.

Question 11

The populations of two competing species, A and B, are modeled by differentiable functions $P_A(t)$ and $P_B(t)$ , respectively, where $t$ is the time in years since the start of an observation. At time $t=5$ years, the population of species A is growing at a rate of 200 individuals per year, while the population of species B is decreasing at a rate of 50 individuals per year. An ecologist uses a weighted total $S(t) = 3P_A(t) - 2P_B(t)$ to measure the overall state of the ecosystem. What is the rate of change of $S(t)$ at $t=5$ years?

150 individuals per year
250 individuals per year
500 individuals per year
700 individuals per year (correct answer)

Explanation: The rate of change of a function is its derivative. We need to find

S'(5)

. First, we translate the given information into mathematical terms. The rate of change of species A at

t=5

P_A'(5) = 200

. Since species B is decreasing, its rate of change is negative:

P_B'(5) = -50

. Next, we find the derivative of the function

S(t) = 3P_A(t) - 2P_B(t)

using the constant multiple and difference rules:

S'(t) = 3P_A'(t) - 2P_B'(t)

. Now, we evaluate this derivative at

t=5

by substituting the given rates:

S'(5) = 3P_A'(5) - 2P_B'(5) = 3(200) - 2(-50) = 600 - (-100) = 600 + 100 = 700

. Thus, the rate of change of

S(t)

t=5

years is 700 individuals per year.

Question 12

Let h(θ) = 2sin(θ) + 4cos(θ). Find the value of h'(π/6). Assume the derivative of sin(θ) is cos(θ) and the derivative of cos(θ) is -sin(θ).

√3 - 2 (correct answer)
1 + 2√3
√3 + 2
1 - 2√3

Explanation: First, find the derivative of h(θ): h'(θ) = 2cos(θ) + 4(-sin(θ)) = 2cos(θ) - 4sin(θ). Next, substitute θ = π/6 into h'(θ): h'(π/6) = 2cos(π/6) - 4sin(π/6). Using the values cos(π/6) = √3/2 and sin(π/6) = 1/2, we get: h'(π/6) = 2(√3/2) - 4(1/2) = √3 - 2.

Question 13

Let f and g be differentiable functions. If h(x) = 5f(x) - 3g(x), and it is known that f'(2) = 4 and g'(2) = -1, what is the instantaneous rate of change of h at x=2?

23 (correct answer)
17
21
7

Explanation: The derivative of h(x) is found using the Constant Multiple and Difference rules: h'(x) = 5f'(x) - 3g'(x). To find the instantaneous rate of change at x=2, we evaluate h'(2): h'(2) = 5f'(2) - 3g'(2) = 5(4) - 3(-1) = 20 + 3 = 23.

Question 14

Let f and g be differentiable functions with f'(1) = -2 and g'(1) = 3. If h(x) = 4g(x) - 2f(x) + π², what is the value of h'(1)?

16 (correct answer)
8
16 + 2π
16 - 2π

Explanation: Using the Sum, Difference, Constant Multiple, and Constant rules, we find the derivative of h(x): h'(x) = 4g'(x) - 2f'(x) + 0. Note that π² is a constant, so its derivative is zero. Evaluating at x=1 gives: h'(1) = 4g'(1) - 2f'(1) = 4(3) - 2(-2) = 12 + 4 = 16.

Question 15

Let f and g be differentiable functions. The function h(x) is defined as h(x) = c⋅f(x) + 2g(x), where c is a constant. The slope of the tangent line to the graph of h(x) at x=a is 5. If f'(a) = -1 and g'(a) = 3, what is the value of c?

-1
1 (correct answer)
11
-11

Explanation: The slope of the tangent line at x=a is h'(a), so h'(a) = 5. Using the derivative rules, h'(x) = c⋅f'(x) + 2g'(x). Substituting the given values: h'(a) = c⋅f'(a) + 2g'(a) becomes 5 = c(-1) + 2(3). This simplifies to 5 = -c + 6. Solving for c gives -1 = -c, so c = 1.

Question 16

Let f(x) be a differentiable function. If g(x) = f(x) + f(a), where a is a constant, which of the following is g'(a)?

f'(a) + f(a)
2f'(a)
f'(a) (correct answer)
0

Explanation: When you encounter a function defined as the sum of terms involving another function, you need to carefully apply differentiation rules to each component. Given

g(x) = f(x) + f(a)

, let's find

g'(a)

by first finding

g'(x)

for any value of

x

, then evaluating at

x = a

. To find

g'(x)

, differentiate each term separately. The derivative of

f(x)

with respect to

x

is simply

f'(x)

. However,

f(a)

is a constant because

a

is a constant—it doesn't depend on

x

at all. The derivative of any constant is zero. Therefore:

g'(x) = f'(x) + 0 = f'(x)

Now evaluating at

x = a

g'(a) = f'(a)

Looking at the incorrect options: Choice A,

f'(a) + f(a)

, mistakenly treats

f(a)

as if it contributes

f(a)

to the derivative rather than zero. This confuses the function value with its derivative. Choice B,

2f'(a)

, incorrectly assumes both terms contribute

f'(a)

to the derivative, forgetting that

f(a)

is constant. Choice D suggests the entire derivative is zero, which would only be true if

g(x)

were entirely constant. The correct answer is C:

f'(a)

. Study tip: Remember that when differentiating, any term that doesn't contain the variable of differentiation is treated as a constant, and constants always differentiate to zero. Focus on identifying which parts of a function actually depend on your variable.

Question 17

Let f, g, and h be differentiable functions. If f'''(x) = 5g''(x) - 4h'(x) + 2, what is the fourth derivative, f⁽⁴⁾(x)?

5g'''(x) - 4h''(x) + 2x
5g'''(x) - 4h''(x) (correct answer)
0
Cannot be determined

Explanation: To find the fourth derivative, f⁽⁴⁾(x), we must differentiate the expression for the third derivative, f'''(x), with respect to x. We apply the Sum, Difference, and Constant Multiple rules to the expression 5g''(x) - 4h'(x) + 2. The derivative is 5g'''(x) - 4h''(x) + 0. Thus, f⁽⁴⁾(x) = 5g'''(x) - 4h''(x).

Question 18

Let L(x) = f(a) + f'(a)(x-a) be the linearization of a differentiable function f(x) at x=a. What is the value of the derivative of the function g(x) = f(x) - L(x) at x=a?

f(a)
f'(a)
0 (correct answer)
1

Explanation: First, find the derivative of g(x). Note that L(x) can be rewritten as L(x) = f'(a)·x + (f(a) - a·f'(a)). In this form, f'(a) and (f(a) - a·f'(a)) are constants. The derivative of L(x) with respect to x is L'(x) = f'(a). Now, find g'(x) using the difference rule: g'(x) = f'(x) - L'(x) = f'(x) - f'(a). To find the value at x=a, we substitute a into this expression: g'(a) = f'(a) - f'(a) = 0.

Question 19

Let f(x) be a differentiable function and let C be a non-zero constant. If h(x) = f(x) + C, what is the relationship between the slope of the tangent line to f(x) at x=a and the slope of the tangent line to h(x) at x=a?

The slope for h(x) is C units greater than the slope for f(x).
The slope for h(x) is C times the slope for f(x).
The slopes are identical. (correct answer)
The slopes are negative reciprocals of each other.

Explanation: The slope of the tangent line at x=a is given by the derivative at that point. The slope for f(x) is f'(a). For h(x), the slope is h'(a). We find the derivative h'(x) using the Sum and Constant rules: h'(x) = f'(x) + 0 = f'(x). Therefore, h'(a) = f'(a) for any value of a. The slopes are identical. Geometrically, adding a constant C to a function shifts its graph vertically, which does not change the slope at any point.

Question 20

Let $f(x)$ and $g(x)$ be differentiable functions. A third function is defined as $h(x) = 3f(x) - 2g(x) + \pi$ . If $f'(2) = 4$ and $g'(2) = -1$ , what is the value of $h'(2)$ ?

10
14 (correct answer)
14 + $\pi$
11

Explanation: To find

h'(x)

, we differentiate

h(x)

term by term using the constant multiple, sum, and difference rules. The derivative of the constant

\pi

is 0. \n

h'(x) = \frac{d}{dx}(3f(x) - 2g(x) + \pi) = 3f'(x) - 2g'(x) + 0

\nNow, we evaluate

h'(x)

x=2

: \n

h'(2) = 3f'(2) - 2g'(2) = 3(4) - 2(-1) = 12 + 2 = 14