Unit 2Linearity of Derivatives

Concept 4.138Built for May 2027 AP Exam

Linearity of Derivatives

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Differentiation is a linear operation, and that single sentence packages three separate facts: sums split, differences split, and constant multiples pass through unchanged. In symbols, $\frac{d}{dx}\bigl[af(x) + bg(x)\bigr] = af'(x) + bg'(x)$ for any constants $a$ and $b$ . The practical value is that a derivative you cannot compute directly can sometimes be assembled from derivatives you were handed: two values from a table, a slope read off a graph, and the coefficients from the question. The practical warning is that linearity is the only structure that survives differentiation unharmed. Products, quotients, and compositions all break, and each of them needs its own rule.

On the AP Exam

What you’ll see: A function defined as a combination of two others, with the pieces supplied by a table, a graph, or a previous part.
What you’ll do: Read each piece's derivative at the target input, multiply by its coefficient, and add, without ever differentiating the combination as a whole.
Memory hook: Coefficients ride, products do not. Addition and scaling survive differentiation; multiplication and composition do not.
Use it on the exam: For MCQs: If the stem hands you $f'(3)$ and $g'(3)$ and asks about $2f - 5g$ , the answer is arithmetic. Options built from $f(3)$ and $g(3)$ are decoys that test whether you can tell values from slopes. For FRQs: Justify the split in one clause, then show the substitution with the numbers visible. The rubric pays for the assembled value, not for a re-derivation of the pieces.

Concept Review

Coefficients Pass Through

Figure 1 — The combination goes in, the differentiated combination comes out, and nothing else about it changes: the 3 stays a 3, the subtraction stays a subtraction, and the −2 stays a −2.

A box holding the combination three f minus two g feeding through a derivative operator into a box holding three f prime minus two g prime, with three notes underneath stating that the coefficients and the operation pass through unchanged

The Statement and Its Two Halves

$\frac{d}{dx}\bigl[af(x) + bg(x)\bigr] = a f'(x) + b g'(x)$

The rule has two independent halves, and it is worth naming them separately. Additivity says the derivative of a sum is the sum of the derivatives. Homogeneity says a constant factor can be moved outside the derivative. Both are inherited from the limit laws, since limits themselves are additive and homogeneous.

Combination	Derivative	Rule used
$f + g$	$f' + g'$	Additivity
$f - g$	$f' - g'$	Additivity with $b = -1$
$4f$	$4f'$	Homogeneity
$4f - 3g$	$4f' - 3g'$	Both
$fg$	not $f'g'$	No linearity available

Assembling a Derivative from Given Pieces

This is where the rule earns its place on the exam. Suppose a table gives $f'(2) = 5$ and $g'(2) = -3$ . Then

$\frac{d}{dx}\bigl[3f(x) - 2g(x)\bigr]\Big|_{x=2} = 3(5) - 2(-3) = 21$

No formula for $f$ or $g$ is needed, and none exists. The combination's derivative is built entirely from values you were handed, which is exactly why the exam supplies tables instead of formulas.

What Linearity Cannot Reach

Expression	What people try	Why it fails
$fg$	$f'g'$	The product changes when both factors change
$\frac{f}{g}$	$\frac{f'}{g'}$	The denominator's change affects the value nonlinearly
$f(g(x))$	$f'(g(x))$	The inner rate multiplies the outer one

Each failure is visible in a one-line counterexample, and each is repaired by a rule in the next three sections. Linearity is the exception, not the pattern.

Common Misconception Check

Myth 1: "If the derivative of $f$ is 5 and of $g$ is 3, then the derivative of $fg$ is 15." Truth: Products need the product rule. The number 15 is meaningless here.
Myth 2: "Linearity means every operation passes through the derivative." Truth: Only addition and scalar multiplication do. Everything else needs its own theorem.
Myth 3: "A table gives me values, so I cannot answer a question about a combination." Truth: If the table gives derivatives, linearity is exactly the tool that turns those entries into the answer.

Topic-Specific Study Method

Split, Scale, Add

Borrowed idea: Linearity is permission. You may split a sum and pull constants out front. Nothing else may be split. A product is not the product of the derivatives. A quotient is not the quotient of them either. When the pieces arrive as a table, this is the rule that lets you combine them into one number. Multiply each derivative by its coefficient, then add.

Stem says	You do	You must not do
$h = 2f - 5g$	$h' = 2f' - 5g'$	Differentiate $f$ and $g$ as a single object
$h = f + 3g$	$h' = f' + 3g'$	Use function values instead of derivatives
$h = \frac{f}{4}$	$h' = \frac{f'}{4}$	Divide the input by 4
$h = fg$	Product rule	Use $f'g'$
$h = f(g(x))$	Chain rule	Use $f'(g(x))$ alone

① Take three random values for $f'(2)$ and $g'(2)$ and compute the derivative of $5f - 2g$ at 2 in one line. Repeat until the multiplication feels automatic.

② Write a function that is a combination of two you already know, differentiate it both as a combination and after expanding it, and confirm the results agree.

③ For each of the three failing rows in the table above, construct a counterexample with simple functions. Keeping those three examples in mind is faster than remembering the warning as a sentence.

Training & Error Diagnosis

Core Archetype (Simplest Form): A table gives $f'(4) = 7$ and $g'(4) = -2$ . Find the derivative of $h(x) = 3f(x) - 5g(x)$ at $x = 4$ .

Standard Execution:

Split. $h'(x) = 3f'(x) - 5g'(x)$ .

Substitute. $h'(4) = 3(7) - 5(-2)$ .

Compute. $21 + 10 = 31$ .

Sanity check. Both terms push upward here, since the second coefficient and the second derivative are both negative.

Exam Variation (Context / Parameter Shift): The number of fish in a lake changes at a rate equal to the entry rate minus the exit rate: $N'(t) = E(t) - L(t)$ , where $E(t) = 20 + 15\sin(\pi t/6)$ and $L(t) = 4 + 2t$ . Determine whether the rate of change of $N$ is increasing or decreasing at $t = 3$ , and justify your answer.

Solution: By linearity, $N''(t) = E'(t) - L'(t)$ , so the question is about the sign of that combination. Here $E'(t) = 15 \cdot \frac{\pi}{6}\cos(\pi t/6)$ and $L'(t) = 2$ , so $N''(3) = 15 \cdot \frac{\pi}{6}\cos(\pi/2) - 2 = 0 - 2 = -2 < 0$ . The rate of change of $N$ is therefore decreasing at $t = 3$ .

Common Traps & Failure Signals:

The product shortcut (Signal: $f'g'$ written for a product): linearity was extended to an operation it does not cover.
The value-for-slope swap (Signal: $f(4)$ used where $f'(4)$ was needed): the table was read on the wrong row.
The coefficient divided (Signal: a coefficient appearing in the denominator of the answer): the multiple was treated as a scaling of the input rather than of the output.

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

Value Swap: You used function values where slopes were required.
Product Assumed $\rightarrow$ Ask whether the operation is addition, scaling, or something else.
Coefficient Misplaced: You multiplied the input by the constant instead of the output, which turns a horizontal stretch into a vertical one and changes every slope in a way the rule never licenses.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Combinations show up inside rate questions and inside splittings of a given function, and the linearity itself is never the named object of a part
10-Year Total Count: 2 of 9 Exams · 2 FRQ Parts; Both parts score a combination of two derivatives, one of them subtracted from the other
FRQ Years Appeared: 2019 (Q1d), 2026 (Q4A)

Practice Examples

MCQ 1

From a Table

A table gives $f'(3) = 5$ and $g'(3) = -2$ . If $h(x) = 4f(x) - 3g(x)$ , what is $h'(3)$ ?

MCQ 2

Which One Is Linear

Which operation can be moved outside a derivative without changing the result?

MCQ 3

Reading the Right Row

A table gives $f(2) = 9$ , $f'(2) = -4$ , $g(2) = 1$ , and $g'(2) = 6$ . If $k(x) = 2f(x) + g(x)$ , what is $k'(2)$ ?

FRQ 1 • A Combination from a Table

The functions $f$ and $g$ are differentiable. Selected values are given in the table.

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
$1$	$4$	$-3$	$7$	$2$
$3$	$5$	$6$	$1$	$-4$

Let $h(x) = 2f(x) - 5g(x)$ and let $k(x) = f(x) + \frac{g(x)}{2}$ .

(a) Find $h'(3)$ . Show the work that leads to your answer.

(b) Find $k'(1)$ .

(c) Explain why linearity cannot produce the derivative of $f(x)g(x)$ at $x = 3$ , then compute that derivative with the rule that does apply.

(d) A classmate claims that because $h = 2f - 5g$ , the value $h(3)$ can be found from the same row. Evaluate that claim.

Cheatsheet

Linearity

	Form
Additivity	$(f + g)' = f' + g'$
Homogeneity	$(cf)' = cf'$
Combined	$(af + bg)' = af' + bg'$
At a point	$(af + bg)'(p) = af'(p) + bg'(p)$
Not linear	$(fg)' \neq f'g'$ , $\left(\frac{f}{g}\right)' \neq \frac{f'}{g'}$

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Unit 2Linearity of Derivatives

Concept 4.138Built for May 2027 AP Exam

Linearity of Derivatives

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A function defined as a combination of two others, with the pieces supplied by a table, a graph, or a previous part.
What you’ll do: Read each piece's derivative at the target input, multiply by its coefficient, and add, without ever differentiating the combination as a whole.
Memory hook: Coefficients ride, products do not. Addition and scaling survive differentiation; multiplication and composition do not.
Use it on the exam: For MCQs: If the stem hands you $f'(3)$ and $g'(3)$ and asks about $2f - 5g$ , the answer is arithmetic. Options built from $f(3)$ and $g(3)$ are decoys that test whether you can tell values from slopes. For FRQs: Justify the split in one clause, then show the substitution with the numbers visible. The rubric pays for the assembled value, not for a re-derivation of the pieces.

Concept Review

Coefficients Pass Through

Figure 1 — The combination goes in, the differentiated combination comes out, and nothing else about it changes: the 3 stays a 3, the subtraction stays a subtraction, and the −2 stays a −2.

The Statement and Its Two Halves

$\frac{d}{dx}\bigl[af(x) + bg(x)\bigr] = a f'(x) + b g'(x)$

Combination	Derivative	Rule used
$f + g$	$f' + g'$	Additivity
$f - g$	$f' - g'$	Additivity with $b = -1$
$4f$	$4f'$	Homogeneity
$4f - 3g$	$4f' - 3g'$	Both
$fg$	not $f'g'$	No linearity available

Assembling a Derivative from Given Pieces

This is where the rule earns its place on the exam. Suppose a table gives $f'(2) = 5$ and $g'(2) = -3$ . Then

$\frac{d}{dx}\bigl[3f(x) - 2g(x)\bigr]\Big|_{x=2} = 3(5) - 2(-3) = 21$

No formula for $f$ or $g$ is needed, and none exists. The combination's derivative is built entirely from values you were handed, which is exactly why the exam supplies tables instead of formulas.

What Linearity Cannot Reach

Expression	What people try	Why it fails
$fg$	$f'g'$	The product changes when both factors change
$\frac{f}{g}$	$\frac{f'}{g'}$	The denominator's change affects the value nonlinearly
$f(g(x))$	$f'(g(x))$	The inner rate multiplies the outer one

Each failure is visible in a one-line counterexample, and each is repaired by a rule in the next three sections. Linearity is the exception, not the pattern.

Common Misconception Check

Myth 1: "If the derivative of $f$ is 5 and of $g$ is 3, then the derivative of $fg$ is 15." Truth: Products need the product rule. The number 15 is meaningless here.
Myth 2: "Linearity means every operation passes through the derivative." Truth: Only addition and scalar multiplication do. Everything else needs its own theorem.
Myth 3: "A table gives me values, so I cannot answer a question about a combination." Truth: If the table gives derivatives, linearity is exactly the tool that turns those entries into the answer.

Topic-Specific Study Method

Split, Scale, Add

Stem says	You do	You must not do
$h = 2f - 5g$	$h' = 2f' - 5g'$	Differentiate $f$ and $g$ as a single object
$h = f + 3g$	$h' = f' + 3g'$	Use function values instead of derivatives
$h = \frac{f}{4}$	$h' = \frac{f'}{4}$	Divide the input by 4
$h = fg$	Product rule	Use $f'g'$
$h = f(g(x))$	Chain rule	Use $f'(g(x))$ alone

① Take three random values for $f'(2)$ and $g'(2)$ and compute the derivative of $5f - 2g$ at 2 in one line. Repeat until the multiplication feels automatic.

② Write a function that is a combination of two you already know, differentiate it both as a combination and after expanding it, and confirm the results agree.

③ For each of the three failing rows in the table above, construct a counterexample with simple functions. Keeping those three examples in mind is faster than remembering the warning as a sentence.

Training & Error Diagnosis

Core Archetype (Simplest Form): A table gives $f'(4) = 7$ and $g'(4) = -2$ . Find the derivative of $h(x) = 3f(x) - 5g(x)$ at $x = 4$ .

Standard Execution:

Split. $h'(x) = 3f'(x) - 5g'(x)$ .

Substitute. $h'(4) = 3(7) - 5(-2)$ .

Compute. $21 + 10 = 31$ .

Sanity check. Both terms push upward here, since the second coefficient and the second derivative are both negative.

Solution: By linearity, $N''(t) = E'(t) - L'(t)$ , so the question is about the sign of that combination. Here $E'(t) = 15 \cdot \frac{\pi}{6}\cos(\pi t/6)$ and $L'(t) = 2$ , so $N''(3) = 15 \cdot \frac{\pi}{6}\cos(\pi/2) - 2 = 0 - 2 = -2 < 0$ . The rate of change of $N$ is therefore decreasing at $t = 3$ .

Common Traps & Failure Signals:

The product shortcut (Signal: $f'g'$ written for a product): linearity was extended to an operation it does not cover.
The value-for-slope swap (Signal: $f(4)$ used where $f'(4)$ was needed): the table was read on the wrong row.
The coefficient divided (Signal: a coefficient appearing in the denominator of the answer): the multiple was treated as a scaling of the input rather than of the output.

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

Value Swap: You used function values where slopes were required.
Product Assumed $\rightarrow$ Ask whether the operation is addition, scaling, or something else.
Coefficient Misplaced: You multiplied the input by the constant instead of the output, which turns a horizontal stretch into a vertical one and changes every slope in a way the rule never licenses.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Combinations show up inside rate questions and inside splittings of a given function, and the linearity itself is never the named object of a part
10-Year Total Count: 2 of 9 Exams · 2 FRQ Parts; Both parts score a combination of two derivatives, one of them subtracted from the other
FRQ Years Appeared: 2019 (Q1d), 2026 (Q4A)

Practice Examples

MCQ 1

From a Table

A table gives $f'(3) = 5$ and $g'(3) = -2$ . If $h(x) = 4f(x) - 3g(x)$ , what is $h'(3)$ ?

MCQ 2

Which One Is Linear

Which operation can be moved outside a derivative without changing the result?

MCQ 3

Reading the Right Row

A table gives $f(2) = 9$ , $f'(2) = -4$ , $g(2) = 1$ , and $g'(2) = 6$ . If $k(x) = 2f(x) + g(x)$ , what is $k'(2)$ ?

FRQ 1 • A Combination from a Table

The functions $f$ and $g$ are differentiable. Selected values are given in the table.

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
$1$	$4$	$-3$	$7$	$2$
$3$	$5$	$6$	$1$	$-4$

Let $h(x) = 2f(x) - 5g(x)$ and let $k(x) = f(x) + \frac{g(x)}{2}$ .

(a) Find $h'(3)$ . Show the work that leads to your answer.

(b) Find $k'(1)$ .

(c) Explain why linearity cannot produce the derivative of $f(x)g(x)$ at $x = 3$ , then compute that derivative with the rule that does apply.

(d) A classmate claims that because $h = 2f - 5g$ , the value $h(3)$ can be found from the same row. Evaluate that claim.

Cheatsheet

Linearity

	Form
Additivity	$(f + g)' = f' + g'$
Homogeneity	$(cf)' = cf'$
Combined	$(af + bg)' = af' + bg'$
At a point	$(af + bg)'(p) = af'(p) + bg'(p)$
Not linear	$(fg)' \neq f'g'$ , $\left(\frac{f}{g}\right)' \neq \frac{f'}{g'}$

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Unit 2Linearity of Derivatives

Concept 4.138Built for May 2027 AP Exam

Linearity of Derivatives

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A function defined as a combination of two others, with the pieces supplied by a table, a graph, or a previous part.
What you’ll do: Read each piece's derivative at the target input, multiply by its coefficient, and add, without ever differentiating the combination as a whole.
Memory hook: Coefficients ride, products do not. Addition and scaling survive differentiation; multiplication and composition do not.
Use it on the exam: For MCQs: If the stem hands you $f'(3)$ and $g'(3)$ and asks about $2f - 5g$ , the answer is arithmetic. Options built from $f(3)$ and $g(3)$ are decoys that test whether you can tell values from slopes. For FRQs: Justify the split in one clause, then show the substitution with the numbers visible. The rubric pays for the assembled value, not for a re-derivation of the pieces.

Concept Review

Coefficients Pass Through

Figure 1 — The combination goes in, the differentiated combination comes out, and nothing else about it changes: the 3 stays a 3, the subtraction stays a subtraction, and the −2 stays a −2.

The Statement and Its Two Halves

$\frac{d}{dx}\bigl[af(x) + bg(x)\bigr] = a f'(x) + b g'(x)$

Combination	Derivative	Rule used
$f + g$	$f' + g'$	Additivity
$f - g$	$f' - g'$	Additivity with $b = -1$
$4f$	$4f'$	Homogeneity
$4f - 3g$	$4f' - 3g'$	Both
$fg$	not $f'g'$	No linearity available

Assembling a Derivative from Given Pieces

This is where the rule earns its place on the exam. Suppose a table gives $f'(2) = 5$ and $g'(2) = -3$ . Then

$\frac{d}{dx}\bigl[3f(x) - 2g(x)\bigr]\Big|_{x=2} = 3(5) - 2(-3) = 21$

No formula for $f$ or $g$ is needed, and none exists. The combination's derivative is built entirely from values you were handed, which is exactly why the exam supplies tables instead of formulas.

What Linearity Cannot Reach

Expression	What people try	Why it fails
$fg$	$f'g'$	The product changes when both factors change
$\frac{f}{g}$	$\frac{f'}{g'}$	The denominator's change affects the value nonlinearly
$f(g(x))$	$f'(g(x))$	The inner rate multiplies the outer one

Each failure is visible in a one-line counterexample, and each is repaired by a rule in the next three sections. Linearity is the exception, not the pattern.

Common Misconception Check

Myth 1: "If the derivative of $f$ is 5 and of $g$ is 3, then the derivative of $fg$ is 15." Truth: Products need the product rule. The number 15 is meaningless here.
Myth 2: "Linearity means every operation passes through the derivative." Truth: Only addition and scalar multiplication do. Everything else needs its own theorem.
Myth 3: "A table gives me values, so I cannot answer a question about a combination." Truth: If the table gives derivatives, linearity is exactly the tool that turns those entries into the answer.

Topic-Specific Study Method

Split, Scale, Add

Stem says	You do	You must not do
$h = 2f - 5g$	$h' = 2f' - 5g'$	Differentiate $f$ and $g$ as a single object
$h = f + 3g$	$h' = f' + 3g'$	Use function values instead of derivatives
$h = \frac{f}{4}$	$h' = \frac{f'}{4}$	Divide the input by 4
$h = fg$	Product rule	Use $f'g'$
$h = f(g(x))$	Chain rule	Use $f'(g(x))$ alone

① Take three random values for $f'(2)$ and $g'(2)$ and compute the derivative of $5f - 2g$ at 2 in one line. Repeat until the multiplication feels automatic.

② Write a function that is a combination of two you already know, differentiate it both as a combination and after expanding it, and confirm the results agree.

③ For each of the three failing rows in the table above, construct a counterexample with simple functions. Keeping those three examples in mind is faster than remembering the warning as a sentence.

Training & Error Diagnosis

Core Archetype (Simplest Form): A table gives $f'(4) = 7$ and $g'(4) = -2$ . Find the derivative of $h(x) = 3f(x) - 5g(x)$ at $x = 4$ .

Standard Execution:

Split. $h'(x) = 3f'(x) - 5g'(x)$ .

Substitute. $h'(4) = 3(7) - 5(-2)$ .

Compute. $21 + 10 = 31$ .

Sanity check. Both terms push upward here, since the second coefficient and the second derivative are both negative.

Solution: By linearity, $N''(t) = E'(t) - L'(t)$ , so the question is about the sign of that combination. Here $E'(t) = 15 \cdot \frac{\pi}{6}\cos(\pi t/6)$ and $L'(t) = 2$ , so $N''(3) = 15 \cdot \frac{\pi}{6}\cos(\pi/2) - 2 = 0 - 2 = -2 < 0$ . The rate of change of $N$ is therefore decreasing at $t = 3$ .

Common Traps & Failure Signals:

The product shortcut (Signal: $f'g'$ written for a product): linearity was extended to an operation it does not cover.
The value-for-slope swap (Signal: $f(4)$ used where $f'(4)$ was needed): the table was read on the wrong row.
The coefficient divided (Signal: a coefficient appearing in the denominator of the answer): the multiple was treated as a scaling of the input rather than of the output.

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

Value Swap: You used function values where slopes were required.
Product Assumed $\rightarrow$ Ask whether the operation is addition, scaling, or something else.
Coefficient Misplaced: You multiplied the input by the constant instead of the output, which turns a horizontal stretch into a vertical one and changes every slope in a way the rule never licenses.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Combinations show up inside rate questions and inside splittings of a given function, and the linearity itself is never the named object of a part
10-Year Total Count: 2 of 9 Exams · 2 FRQ Parts; Both parts score a combination of two derivatives, one of them subtracted from the other
FRQ Years Appeared: 2019 (Q1d), 2026 (Q4A)

Practice Examples

MCQ 1

From a Table

A table gives $f'(3) = 5$ and $g'(3) = -2$ . If $h(x) = 4f(x) - 3g(x)$ , what is $h'(3)$ ?

MCQ 2

Which One Is Linear

Which operation can be moved outside a derivative without changing the result?

MCQ 3

Reading the Right Row

A table gives $f(2) = 9$ , $f'(2) = -4$ , $g(2) = 1$ , and $g'(2) = 6$ . If $k(x) = 2f(x) + g(x)$ , what is $k'(2)$ ?

FRQ 1 • A Combination from a Table

The functions $f$ and $g$ are differentiable. Selected values are given in the table.

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
$1$	$4$	$-3$	$7$	$2$
$3$	$5$	$6$	$1$	$-4$

Let $h(x) = 2f(x) - 5g(x)$ and let $k(x) = f(x) + \frac{g(x)}{2}$ .

(a) Find $h'(3)$ . Show the work that leads to your answer.

(b) Find $k'(1)$ .

(c) Explain why linearity cannot produce the derivative of $f(x)g(x)$ at $x = 3$ , then compute that derivative with the rule that does apply.

(d) A classmate claims that because $h = 2f - 5g$ , the value $h(3)$ can be found from the same row. Evaluate that claim.

Cheatsheet

Linearity

	Form
Additivity	$(f + g)' = f' + g'$
Homogeneity	$(cf)' = cf'$
Combined	$(af + bg)' = af' + bg'$
At a point	$(af + bg)'(p) = af'(p) + bg'(p)$
Not linear	$(fg)' \neq f'g'$ , $\left(\frac{f}{g}\right)' \neq \frac{f'}{g'}$

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