Unit 2Basic Rules

Concept 4.140Built for May 2027 AP Exam

Basic Rules

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Three rules do most of the work in this unit, and none of them requires a limit once the power rule is in hand. The derivative of a constant is zero, because a horizontal line has no slope. A constant multiple passes through the derivative unchanged, because scaling the output scales every slope by the same factor. A sum or difference differentiates term by term, because the limit of a sum is the sum of the limits. Those three, plus the power rule, differentiate every polynomial in one line. What they do not do is equally important: there is no rule that lets you differentiate a product, a quotient, or a composition by handling the pieces separately, and those three cases arrive in the next sections.

On the AP Exam

What you’ll see: A polynomial, a sum of a polynomial and a constant, or a term with a coefficient, inside a rate, a tangent, or a second derivative.
What you’ll do: Differentiate term by term, keep every coefficient, and let every standalone constant disappear.
Memory hook: Constants out, constants gone. A multiplier survives the derivative; an additive constant does not.
Use it on the exam: For MCQs: The distractors usually keep an additive constant or drop a coefficient. Count the terms in your derivative against the terms in the original: only the constant term should vanish. For FRQs: Speed matters here, because the derivative is a step rather than an answer. Write the derivative in one pass, then spend the space on the substitution and the interpretation the part is paying for.

Concept Review

Three Rules in One Pass

Figure 1 — Each card states one rule and applies it immediately. The rules act on single blocks, which is why a polynomial can be differentiated left to right without regrouping anything.

Three stacked rule cards for the constant rule, the constant multiple rule, and the sum and difference rule, each paired with a worked example and a one-line note

Why They Hold

Each rule is a limit statement, and each follows from the limit laws:

$\frac{d}{dx}[c] = \lim_{h\to 0}\frac{c-c}{h} = 0, \qquad \frac{d}{dx}[cf] = c\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = cf'(x)$

$\frac{d}{dx}[f \pm g] = \lim_{h\to 0}\frac{[f(x+h)-f(x)] \pm [g(x+h)-g(x)]}{h} = f'(x) \pm g'(x)$

Nothing in these arguments is specific to polynomials. They hold for every differentiable function, which is why the rules keep working when the pieces become trigonometric or exponential.

What the Rules Do Not Cover

You might hope	The truth	Where it is handled
$(fg)' = f'g'$	False in general	Product rule
$\left(\frac{f}{g}\right)' = \frac{f'}{g'}$	False in general	Quotient rule
$(f \circ g)' = f' \circ g'$	False in general	Chain rule

A single counterexample settles each row. With $f(x) = x$ and $g(x) = x$ , the product is $x^2$ , whose derivative is $2x$ , while $f'g' = 1$ . The failure is not a technicality; it is the reason the next three sections exist.

Reading a Derivative Backwards

Because coefficients pass through, you can often reconstruct the original from the derivative. If $f'(x) = 6x^2 - 4$ , then $f$ must contain $2x^3 - 4x$ plus some constant, since differentiating $2x^3$ gives $6x^2$ and differentiating $-4x$ gives $-4$ . That reconstruction is antidifferentiation, and it works for exactly the same reason the rules do.

Common Misconception Check

Myth 1: "The derivative of $5x^3$ is $5x^2$ ." Truth: The power rule still multiplies by the exponent, so the answer is $15x^2$ . The constant multiple only preserves the factor of 5.
Myth 2: "A constant term becomes 1." Truth: It becomes 0. The value 1 is what the derivative of $x$ gives, not what the derivative of $7$ gives.
Myth 3: "Sum rules let you split a product." Truth: Sums split; products do not. Confusing the two produces the most common structural error in this unit.

Topic-Specific Study Method

One Pass, Left to Right

Borrowed idea: Work from left to right, and treat each block as though the others were absent. A coefficient stays exactly where it is while its block is differentiated. A constant standing alone disappears, since it adds nothing to any slope. A subtraction sign belongs to the block that follows it, and a dropped sign stays invisible until the substitution step. When the expression is long, cover the other terms and differentiate what you can see. Then count the terms in your answer. Only the constant should have gone missing, and any other gap means a block was skipped.

Given	Derivative	What the rules did
$7$	$0$	Constant rule
$7x^2$	$14x$	Multiple plus power
$7x^2 - 7$	$14x$	Constant rule removed the tail
$\frac{x^3}{4}$	$\frac{3x^2}{4}$	Multiple $\frac{1}{4}$ passed through
$2x - x^2$	$2 - 2x$	Difference, sign attached to the second block

① Differentiate five polynomials with a constant term, then count the terms in each answer. The count should drop by exactly one, and that check catches most slips faster than re-deriving.

② Write a function whose derivative is $4x^3 - 6x + 1$ , then check it by differentiating back. Reconstructing forces you to see the rules in reverse.

③ Take a product such as $x \cdot x$ and differentiate it two ways, once correctly and once by the false rule $(fg)' = f'g'$ . The difference between $2x$ and $1$ is the reason the next section exists.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = 3x^4 - 2x^2 + 5x - 9$ .

Standard Execution:

Term by term. $3x^4 \to 12x^3$ , $-2x^2 \to -4x$ , $5x \to 5$ , $-9 \to 0$ .

Assemble. $f'(x) = 12x^3 - 4x + 5$ .

Count. Four terms in, three out, and the missing one is the constant.

Check a value. At $x = 1$ , $f' = 13$ , and the graph of $f$ rises steeply there, which matches.

Exam Variation (Context / Parameter Shift): A quantity is modeled by $Q(t) = \frac{t^4}{4} - 3t^2 + 8$ for $t \ge 0$ . Find the rate of change at $t = 2$ , with units of $Q$ per unit of $t$ , and determine whether $Q$ is increasing or decreasing there.

Solution: The coefficient $\frac{1}{4}$ passes through: $Q'(t) = t^3 - 6t$ . At $t = 2$ this gives $8 - 12 = -4$ units of $Q$ per unit of $t$ . The negative sign means $Q$ is decreasing at that instant, and the size says it is falling four units per unit of time.

Common Traps & Failure Signals:

The surviving constant (Signal: an extra term in the derivative): the constant rule was skipped.
The swallowed coefficient (Signal: a derivative exactly the size of the power rule's output without the multiplier): the constant multiple was divided out instead of carried.
The sign drift (Signal: a plus where the original had a minus): the subtraction was not attached to the following block.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Constant Kept: You carried an additive constant into the derivative.
Coefficient Lost $\rightarrow$ Say the multiplier out loud before differentiating the block.
Rules Blended: You applied the sum rule to a product or a quotient, which happens when the three rules are memorised as slogans rather than as statements about separate blocks.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; The rules are the grammar of every derivative rather than an examined object, and released BC parts usually supply the derivative they intend you to use
10-Year Total Count: 4 of 9 Exams · 4 FRQ Parts; Counted where a constant, a constant multiple, or a sum survives all the way into the reported derivative
FRQ Years Appeared: 2018 (Q2a), 2019 (Q1d), 2023 (Q1d), 2026 (Q4A)

Practice Examples

MCQ 1

A Constant Term

If $f(x) = 4x^3 - 6x + 11$ , what is $f'(x)$ ?

MCQ 2

A Coefficient

If $g(x) = \frac{x^5}{2}$ , what is $g'(x)$ ?

MCQ 3

What the Rules Cannot Do

Which statement is true for differentiable functions $f$ and $g$ ?

FRQ 1 • A Quantity Model

A quantity is modeled by $Q(t) = \frac{t^4}{4} - 3t^2 + 8$ for $0 \le t \le 4$ , where $Q$ is measured in liters and $t$ in minutes.

(a) Find $Q'(t)$ .

(b) Find the rate of change of $Q$ at $t = 2$ minutes, with units, and state whether $Q$ is increasing or decreasing there.

(c) Find every time in $0 < t < 4$ at which $Q'(t) = 0$ .

(d) Explain why the constant $8$ does not appear in $Q'(t)$ , and state what the value of $Q$ at $t = 0$ contributes to the rate of change.

Cheatsheet

Basic Rules

	Statement	Example
Constant	$\frac{d}{dx}[c] = 0$	$\frac{d}{dx}[9] = 0$
Constant multiple	$\frac{d}{dx}[cf] = cf'$	$\frac{d}{dx}[5x^3] = 15x^2$
Sum	$\frac{d}{dx}[f+g] = f'+g'$	$\frac{d}{dx}[x^2 + x] = 2x + 1$
Difference	$\frac{d}{dx}[f-g] = f'-g'$	$\frac{d}{dx}[x^2 - x] = 2x - 1$
False product	$(fg)' \neq f'g'$	$x \cdot x$ gives $2x$ , not $1$

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Unit 2Basic Rules

Concept 4.140Built for May 2027 AP Exam

Basic Rules

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A polynomial, a sum of a polynomial and a constant, or a term with a coefficient, inside a rate, a tangent, or a second derivative.
What you’ll do: Differentiate term by term, keep every coefficient, and let every standalone constant disappear.
Memory hook: Constants out, constants gone. A multiplier survives the derivative; an additive constant does not.
Use it on the exam: For MCQs: The distractors usually keep an additive constant or drop a coefficient. Count the terms in your derivative against the terms in the original: only the constant term should vanish. For FRQs: Speed matters here, because the derivative is a step rather than an answer. Write the derivative in one pass, then spend the space on the substitution and the interpretation the part is paying for.

Concept Review

Three Rules in One Pass

Figure 1 — Each card states one rule and applies it immediately. The rules act on single blocks, which is why a polynomial can be differentiated left to right without regrouping anything.

Three stacked rule cards for the constant rule, the constant multiple rule, and the sum and difference rule, each paired with a worked example and a one-line note

Why They Hold

Each rule is a limit statement, and each follows from the limit laws:

$\frac{d}{dx}[c] = \lim_{h\to 0}\frac{c-c}{h} = 0, \qquad \frac{d}{dx}[cf] = c\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = cf'(x)$

$\frac{d}{dx}[f \pm g] = \lim_{h\to 0}\frac{[f(x+h)-f(x)] \pm [g(x+h)-g(x)]}{h} = f'(x) \pm g'(x)$

Nothing in these arguments is specific to polynomials. They hold for every differentiable function, which is why the rules keep working when the pieces become trigonometric or exponential.

What the Rules Do Not Cover

You might hope	The truth	Where it is handled
$(fg)' = f'g'$	False in general	Product rule
$\left(\frac{f}{g}\right)' = \frac{f'}{g'}$	False in general	Quotient rule
$(f \circ g)' = f' \circ g'$	False in general	Chain rule

Reading a Derivative Backwards

Common Misconception Check

Myth 1: "The derivative of $5x^3$ is $5x^2$ ." Truth: The power rule still multiplies by the exponent, so the answer is $15x^2$ . The constant multiple only preserves the factor of 5.
Myth 2: "A constant term becomes 1." Truth: It becomes 0. The value 1 is what the derivative of $x$ gives, not what the derivative of $7$ gives.
Myth 3: "Sum rules let you split a product." Truth: Sums split; products do not. Confusing the two produces the most common structural error in this unit.

Topic-Specific Study Method

One Pass, Left to Right

Given	Derivative	What the rules did
$7$	$0$	Constant rule
$7x^2$	$14x$	Multiple plus power
$7x^2 - 7$	$14x$	Constant rule removed the tail
$\frac{x^3}{4}$	$\frac{3x^2}{4}$	Multiple $\frac{1}{4}$ passed through
$2x - x^2$	$2 - 2x$	Difference, sign attached to the second block

① Differentiate five polynomials with a constant term, then count the terms in each answer. The count should drop by exactly one, and that check catches most slips faster than re-deriving.

② Write a function whose derivative is $4x^3 - 6x + 1$ , then check it by differentiating back. Reconstructing forces you to see the rules in reverse.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = 3x^4 - 2x^2 + 5x - 9$ .

Standard Execution:

Term by term. $3x^4 \to 12x^3$ , $-2x^2 \to -4x$ , $5x \to 5$ , $-9 \to 0$ .

Assemble. $f'(x) = 12x^3 - 4x + 5$ .

Count. Four terms in, three out, and the missing one is the constant.

Check a value. At $x = 1$ , $f' = 13$ , and the graph of $f$ rises steeply there, which matches.

Solution: The coefficient $\frac{1}{4}$ passes through: $Q'(t) = t^3 - 6t$ . At $t = 2$ this gives $8 - 12 = -4$ units of $Q$ per unit of $t$ . The negative sign means $Q$ is decreasing at that instant, and the size says it is falling four units per unit of time.

Common Traps & Failure Signals:

The surviving constant (Signal: an extra term in the derivative): the constant rule was skipped.
The swallowed coefficient (Signal: a derivative exactly the size of the power rule's output without the multiplier): the constant multiple was divided out instead of carried.
The sign drift (Signal: a plus where the original had a minus): the subtraction was not attached to the following block.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Constant Kept: You carried an additive constant into the derivative.
Coefficient Lost $\rightarrow$ Say the multiplier out loud before differentiating the block.
Rules Blended: You applied the sum rule to a product or a quotient, which happens when the three rules are memorised as slogans rather than as statements about separate blocks.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; The rules are the grammar of every derivative rather than an examined object, and released BC parts usually supply the derivative they intend you to use
10-Year Total Count: 4 of 9 Exams · 4 FRQ Parts; Counted where a constant, a constant multiple, or a sum survives all the way into the reported derivative
FRQ Years Appeared: 2018 (Q2a), 2019 (Q1d), 2023 (Q1d), 2026 (Q4A)

Practice Examples

MCQ 1

A Constant Term

If $f(x) = 4x^3 - 6x + 11$ , what is $f'(x)$ ?

MCQ 2

A Coefficient

If $g(x) = \frac{x^5}{2}$ , what is $g'(x)$ ?

MCQ 3

What the Rules Cannot Do

Which statement is true for differentiable functions $f$ and $g$ ?

FRQ 1 • A Quantity Model

A quantity is modeled by $Q(t) = \frac{t^4}{4} - 3t^2 + 8$ for $0 \le t \le 4$ , where $Q$ is measured in liters and $t$ in minutes.

(a) Find $Q'(t)$ .

(b) Find the rate of change of $Q$ at $t = 2$ minutes, with units, and state whether $Q$ is increasing or decreasing there.

(c) Find every time in $0 < t < 4$ at which $Q'(t) = 0$ .

(d) Explain why the constant $8$ does not appear in $Q'(t)$ , and state what the value of $Q$ at $t = 0$ contributes to the rate of change.

Cheatsheet

Basic Rules

	Statement	Example
Constant	$\frac{d}{dx}[c] = 0$	$\frac{d}{dx}[9] = 0$
Constant multiple	$\frac{d}{dx}[cf] = cf'$	$\frac{d}{dx}[5x^3] = 15x^2$
Sum	$\frac{d}{dx}[f+g] = f'+g'$	$\frac{d}{dx}[x^2 + x] = 2x + 1$
Difference	$\frac{d}{dx}[f-g] = f'-g'$	$\frac{d}{dx}[x^2 - x] = 2x - 1$
False product	$(fg)' \neq f'g'$	$x \cdot x$ gives $2x$ , not $1$

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Unit 2Basic Rules

Concept 4.140Built for May 2027 AP Exam

Basic Rules

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A polynomial, a sum of a polynomial and a constant, or a term with a coefficient, inside a rate, a tangent, or a second derivative.
What you’ll do: Differentiate term by term, keep every coefficient, and let every standalone constant disappear.
Memory hook: Constants out, constants gone. A multiplier survives the derivative; an additive constant does not.
Use it on the exam: For MCQs: The distractors usually keep an additive constant or drop a coefficient. Count the terms in your derivative against the terms in the original: only the constant term should vanish. For FRQs: Speed matters here, because the derivative is a step rather than an answer. Write the derivative in one pass, then spend the space on the substitution and the interpretation the part is paying for.

Concept Review

Three Rules in One Pass

Figure 1 — Each card states one rule and applies it immediately. The rules act on single blocks, which is why a polynomial can be differentiated left to right without regrouping anything.

Three stacked rule cards for the constant rule, the constant multiple rule, and the sum and difference rule, each paired with a worked example and a one-line note

Why They Hold

Each rule is a limit statement, and each follows from the limit laws:

$\frac{d}{dx}[c] = \lim_{h\to 0}\frac{c-c}{h} = 0, \qquad \frac{d}{dx}[cf] = c\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = cf'(x)$

$\frac{d}{dx}[f \pm g] = \lim_{h\to 0}\frac{[f(x+h)-f(x)] \pm [g(x+h)-g(x)]}{h} = f'(x) \pm g'(x)$

Nothing in these arguments is specific to polynomials. They hold for every differentiable function, which is why the rules keep working when the pieces become trigonometric or exponential.

What the Rules Do Not Cover

You might hope	The truth	Where it is handled
$(fg)' = f'g'$	False in general	Product rule
$\left(\frac{f}{g}\right)' = \frac{f'}{g'}$	False in general	Quotient rule
$(f \circ g)' = f' \circ g'$	False in general	Chain rule

Reading a Derivative Backwards

Common Misconception Check

Myth 1: "The derivative of $5x^3$ is $5x^2$ ." Truth: The power rule still multiplies by the exponent, so the answer is $15x^2$ . The constant multiple only preserves the factor of 5.
Myth 2: "A constant term becomes 1." Truth: It becomes 0. The value 1 is what the derivative of $x$ gives, not what the derivative of $7$ gives.
Myth 3: "Sum rules let you split a product." Truth: Sums split; products do not. Confusing the two produces the most common structural error in this unit.

Topic-Specific Study Method

One Pass, Left to Right

Given	Derivative	What the rules did
$7$	$0$	Constant rule
$7x^2$	$14x$	Multiple plus power
$7x^2 - 7$	$14x$	Constant rule removed the tail
$\frac{x^3}{4}$	$\frac{3x^2}{4}$	Multiple $\frac{1}{4}$ passed through
$2x - x^2$	$2 - 2x$	Difference, sign attached to the second block

① Differentiate five polynomials with a constant term, then count the terms in each answer. The count should drop by exactly one, and that check catches most slips faster than re-deriving.

② Write a function whose derivative is $4x^3 - 6x + 1$ , then check it by differentiating back. Reconstructing forces you to see the rules in reverse.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = 3x^4 - 2x^2 + 5x - 9$ .

Standard Execution:

Term by term. $3x^4 \to 12x^3$ , $-2x^2 \to -4x$ , $5x \to 5$ , $-9 \to 0$ .

Assemble. $f'(x) = 12x^3 - 4x + 5$ .

Count. Four terms in, three out, and the missing one is the constant.

Check a value. At $x = 1$ , $f' = 13$ , and the graph of $f$ rises steeply there, which matches.

Solution: The coefficient $\frac{1}{4}$ passes through: $Q'(t) = t^3 - 6t$ . At $t = 2$ this gives $8 - 12 = -4$ units of $Q$ per unit of $t$ . The negative sign means $Q$ is decreasing at that instant, and the size says it is falling four units per unit of time.

Common Traps & Failure Signals:

The surviving constant (Signal: an extra term in the derivative): the constant rule was skipped.
The swallowed coefficient (Signal: a derivative exactly the size of the power rule's output without the multiplier): the constant multiple was divided out instead of carried.
The sign drift (Signal: a plus where the original had a minus): the subtraction was not attached to the following block.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Constant Kept: You carried an additive constant into the derivative.
Coefficient Lost $\rightarrow$ Say the multiplier out loud before differentiating the block.
Rules Blended: You applied the sum rule to a product or a quotient, which happens when the three rules are memorised as slogans rather than as statements about separate blocks.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; The rules are the grammar of every derivative rather than an examined object, and released BC parts usually supply the derivative they intend you to use
10-Year Total Count: 4 of 9 Exams · 4 FRQ Parts; Counted where a constant, a constant multiple, or a sum survives all the way into the reported derivative
FRQ Years Appeared: 2018 (Q2a), 2019 (Q1d), 2023 (Q1d), 2026 (Q4A)

Practice Examples

MCQ 1

A Constant Term

If $f(x) = 4x^3 - 6x + 11$ , what is $f'(x)$ ?

MCQ 2

A Coefficient

If $g(x) = \frac{x^5}{2}$ , what is $g'(x)$ ?

MCQ 3

What the Rules Cannot Do

Which statement is true for differentiable functions $f$ and $g$ ?

FRQ 1 • A Quantity Model

A quantity is modeled by $Q(t) = \frac{t^4}{4} - 3t^2 + 8$ for $0 \le t \le 4$ , where $Q$ is measured in liters and $t$ in minutes.

(a) Find $Q'(t)$ .

(b) Find the rate of change of $Q$ at $t = 2$ minutes, with units, and state whether $Q$ is increasing or decreasing there.

(c) Find every time in $0 < t < 4$ at which $Q'(t) = 0$ .

(d) Explain why the constant $8$ does not appear in $Q'(t)$ , and state what the value of $Q$ at $t = 0$ contributes to the rate of change.

Cheatsheet

Basic Rules

	Statement	Example
Constant	$\frac{d}{dx}[c] = 0$	$\frac{d}{dx}[9] = 0$
Constant multiple	$\frac{d}{dx}[cf] = cf'$	$\frac{d}{dx}[5x^3] = 15x^2$
Sum	$\frac{d}{dx}[f+g] = f'+g'$	$\frac{d}{dx}[x^2 + x] = 2x + 1$
Difference	$\frac{d}{dx}[f-g] = f'-g'$	$\frac{d}{dx}[x^2 - x] = 2x - 1$
False product	$(fg)' \neq f'g'$	$x \cdot x$ gives $2x$ , not $1$

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