Unit 3The Chain Rule Basics

Concept 4.129Built for May 2027 AP Exam

The Chain Rule Basics

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The chain rule is what keeps every other derivative rule alive inside a composite function. Differentiate the outside, leave the inside exactly as it stood, then multiply by the derivative of the inside. For $f(g(x))$ the result is $f'(g(x)) \cdot g'(x)$ , and that second factor is a multiplication, never an addition. The rule also stacks, because a function nested three deep hands you three factors in a chain. Almost every lost point on this topic is a second factor that never got written down, not a rule that was known incorrectly. The fix is procedural rather than conceptual: name the inside before you differentiate anything, then write the second factor before simplifying the first.

On the AP Exam

The chain rule is rarely a question by itself. It is the step that makes somebody else's question possible, and it sits inside the first line of tangent-line, rate-of-change, and implicit work. A missing factor there does not stay local; it changes every number that follows it.

What you’ll see: A sine, an exponential, a logarithm, or a power whose argument is not a bare $x$ , together with a request for a derivative, a slope, or an instantaneous rate.
What you’ll do: Name the outside and the inside, differentiate the outside at the untouched inside, then multiply by the inside derivative.
Memory hook: Outside first, inside stays, times inside. The inner derivative is a factor you multiply by, so it can never turn into a plus sign.
Use it on the exam: For MCQs: Every distractor family is built from the same break. The option that omits the inner derivative is always present, so check the number of factors in your answer before you check the algebra. For FRQs: When a part asks for $\frac{d}{dx}\left[e^{f(x)}\right]$ or $\frac{d}{dt}\left[\sin(g(t))\right]$ , the justification line must exhibit the inner factor. An answer that only differentiates the outside earns the form point and loses the reasoning point.

Concept Review

Two Factors, One Motion

Figure 1 — The chain rule produces two factors, not one. The outside function is differentiated at the untouched inside, and the derivative of the inside arrives as a multiplicand.

OUTER DERIVATIVE TIMES INNER DERIVATIVE

Reading the Nest Before You Differentiate

The only decision the rule requires is which layer is outside. Everything after that is bookkeeping.

Function	Outside $f$	Inside $g$	Derivative
$\sin 3x$	$\sin u$	$3x$	$3\cos 3x$
$e^{x^2}$	$e^u$	$x^2$	$2x\,e^{x^2}$
$(2x+1)^5$	$u^5$	$2x+1$	$10(2x+1)^4$
$\ln(x^2+1)$	$\ln u$	$x^2+1$	$\dfrac{2x}{x^2+1}$
$\sqrt{4x-3}$	$u^{1/2}$	$4x-3$	$\dfrac{2}{\sqrt{4x-3}}$

The outside is the function whose argument is the whole inside expression. In $\sin^2 x$ the outside is the squaring and the inside is $\sin x$ ; in $2^{\sin x}$ the outside is the exponential and the inside is $\sin x$ . Reading the expression aloud with the phrase "of" inserted at the right place settles the question in one pass.

The Four Moves, in Order

Name the inside. Write $g(x) =$ the expression sitting in the argument, and confirm the original really is $f(g(x))$ .
Differentiate the outside at the untouched inside. For $\sin 3x$ that is $\cos 3x$ , not $\cos x$ .
Multiply by the inside derivative. This factor is what the exam is testing, and it is the one most often absent.
Simplify only afterwards. Cancelling before the second factor is written is how the factor gets lost.

Common Misconception Check

Myth 1: "The chain rule applies only when the function looks like $f(g(x))$ written out." Truth: It applies whenever the argument of a function is anything other than a bare $x$ , including $e^{2x}$ , $\ln 5x$ , and $\sqrt{x+1}$ .
Myth 2: "The inner derivative is added to the outside derivative." Truth: The two pieces are multiplied, so a minus sign in the outside derivative survives and an addition never appears.
Myth 3: "A constant inside needs no factor." Truth: A constant inside contributes a constant factor, so $\sin 3x$ picks up a $3$ and $\sin(x + 1)$ picks up a $1$ that is invisible but present.

Topic-Specific Study Method

Name the Inside, Then Differentiate

Borrowed idea: The chain rule is one sentence, and it applies to every nested function you will meet. Differentiate the outside first, and leave the inside expression exactly as it was. Then multiply that result by the derivative of the inside, which is the second factor. Those two factors never merge into one, and neither of them may be dropped. Nearly every lost point in this topic is a second factor that never got written down. The rule stacks cleanly, because every additional layer contributes one more factor to the product.

Exam disguise	What you see on the page	What your brain must translate	Required execution
A linear inside	$\sin 5x$ , $e^{3x}$ , $\sqrt{2x+1}$	The argument carries a constant slope	Multiply by that constant, even when it is $1$
A power inside	$\sin(x^2)$ , $e^{x^2}$	The inside is not linear	Multiply by $2x$ , not by $2$
A function inside	$\sin(\cos x)$ , $e^{\sin x}$	Two layers, so two factors	Differentiate outward, multiplying at each layer
A power of a function	$\sin^2 x$ , $(x^2+1)^3$	The exponent is outside, the function is inside	Bring the power down, keep the inside, then multiply

① Write $f(x) = \sin(3x)$ and differentiate it three times in a row, checking that a factor of $3$ appears each time. The rule is the same motion repeated, and the repetition is what makes it stick.

② Take five expressions from your own homework that contain nested parentheses, and for each one write only the inside function, without differentiating. Naming the inside is the whole decision, and it takes about thirty seconds per expression.

③ On any tangent-line question, differentiate completely first and substitute second. Substituting into a half-differentiated expression silently discards the second factor, and the number you get afterwards looks entirely plausible.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\frac{d}{dx}\left[\sin(3x) + e^{x^2}\right]$ .

Standard Execution:

Split. The sum rule lets each term be handled alone.

First term. The outside is sine, the inside is $3x$ , so the derivative is $3\cos 3x$ .

Second term. The outside is the exponential, the inside is $x^2$ , so the derivative is $2x\,e^{x^2}$ .

Assemble. The derivative is $3\cos 3x + 2x\,e^{x^2}$ .

Stop. No simplification is possible, and none is required.

Exam Variation (Context / Parameter Shift): Let $f(x) = (2x+1)^5$ . Find the equation of the line tangent to the graph of $f$ at $x = 1$ .

Solution: The chain rule gives $f'(x) = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4$ . At $x = 1$ the point is $(1, 243)$ and the slope is $f'(1) = 10 \cdot 81 = 810$ . The tangent line is $y - 243 = 810(x - 1)$ . The factor of $2$ inside the power is the entire difference between $810$ and the $405$ that a missing chain rule would produce.

Common Traps & Failure Signals:

The single factor (Signal: an answer with one multiplicative piece): the inside was differentiated mentally and then dropped instead of written.
The wrong evaluation point (Signal: $\cos x$ where the page had $\cos 3x$ ): the outside was differentiated at $x$ rather than at the untouched inside.
The early cancellation (Signal: a factor that disappears between two lines): simplification ran ahead of the second factor, and the algebra quietly absorbed it.

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

Inner Drop $\rightarrow$ Write the second factor first.
Layer Confusion: You named the outside function as the inside one, so the first factor in your answer was evaluated at the wrong expression and the remaining factor had nothing left to multiply.
Flat Reading: You saw a single function where the page had two, so the answer came out with one factor instead of two.

Past 10-Year AP Exam Appearance

Testing Frequency: Every Single Year; Every released exam from 2017 to 2026 contains at least one scored part whose answer cannot be reached without differentiating a composite
10-Year Total Count: 9 of 9 Exams · 21 FRQ Parts; Twenty-one parts across the nine released sets, counted part by part; the chain is scored through the derivative line rather than through a named part
FRQ Years Appeared: 2017 (Q5a, Q6a, Q6b, Q6c), 2018 (Q2a), 2019 (Q1d, Q6b, Q6c), 2021 (Q2c), 2022 (Q1c, Q2b), 2023 (Q1d, Q2b, Q5a, Q5b, Q6d), 2024 (Q2b, Q5d), 2025 (Q5A, Q6D), 2026 (Q6B)

Practice Examples

MCQ 1

The Linear Inside

If $f(x) = \sin(3x)$ , what is $f'(x)$ ?

MCQ 2

A Power Around a Linear

Let $g(x) = (2x+1)^5$ . What is $g'(0)$ ?

MCQ 3

The Exponential Trap

What is the derivative of $h(x) = e^{x^2}$ ?

FRQ 4

A Radical and a Tangent

Let $f$ be the function defined by $f(x) = \sqrt{3x + 1}$ for $x \ge -\frac{1}{3}$ .

(a) Find $f'(x)$ .

(b) Find the equation of the line tangent to the graph of $f$ at $x = 1$ .

(c) The function $g$ is defined by $g(x) = f(x^2)$ . Find $g'(x)$ and evaluate $g'(1)$ .

(d) A classmate writes $f'(x) = \frac{1}{2\sqrt{3x+1}}$ and claims the answer is complete. Identify the missing piece and explain what it changes about the slope at $x = 1$ .

Cheatsheet

The Chain Rule

	Move
Argument is a bare $x$	No chain rule needed
Argument is $ax + b$	Multiply by $a$
Argument is a function of $x$	Multiply by its derivative
Two layers	Two factors, one per layer
Three layers	Three factors, one per layer
Inside is a constant	The derivative is zero, since the function is constant

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Unit 3The Chain Rule Basics

Concept 4.129Built for May 2027 AP Exam

The Chain Rule Basics

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A sine, an exponential, a logarithm, or a power whose argument is not a bare $x$ , together with a request for a derivative, a slope, or an instantaneous rate.
What you’ll do: Name the outside and the inside, differentiate the outside at the untouched inside, then multiply by the inside derivative.
Memory hook: Outside first, inside stays, times inside. The inner derivative is a factor you multiply by, so it can never turn into a plus sign.
Use it on the exam: For MCQs: Every distractor family is built from the same break. The option that omits the inner derivative is always present, so check the number of factors in your answer before you check the algebra. For FRQs: When a part asks for $\frac{d}{dx}\left[e^{f(x)}\right]$ or $\frac{d}{dt}\left[\sin(g(t))\right]$ , the justification line must exhibit the inner factor. An answer that only differentiates the outside earns the form point and loses the reasoning point.

Concept Review

Two Factors, One Motion

Figure 1 — The chain rule produces two factors, not one. The outside function is differentiated at the untouched inside, and the derivative of the inside arrives as a multiplicand.

OUTER DERIVATIVE TIMES INNER DERIVATIVE

Reading the Nest Before You Differentiate

The only decision the rule requires is which layer is outside. Everything after that is bookkeeping.

Function	Outside $f$	Inside $g$	Derivative
$\sin 3x$	$\sin u$	$3x$	$3\cos 3x$
$e^{x^2}$	$e^u$	$x^2$	$2x\,e^{x^2}$
$(2x+1)^5$	$u^5$	$2x+1$	$10(2x+1)^4$
$\ln(x^2+1)$	$\ln u$	$x^2+1$	$\dfrac{2x}{x^2+1}$
$\sqrt{4x-3}$	$u^{1/2}$	$4x-3$	$\dfrac{2}{\sqrt{4x-3}}$

The Four Moves, in Order

Name the inside. Write $g(x) =$ the expression sitting in the argument, and confirm the original really is $f(g(x))$ .
Differentiate the outside at the untouched inside. For $\sin 3x$ that is $\cos 3x$ , not $\cos x$ .
Multiply by the inside derivative. This factor is what the exam is testing, and it is the one most often absent.
Simplify only afterwards. Cancelling before the second factor is written is how the factor gets lost.

Common Misconception Check

Myth 1: "The chain rule applies only when the function looks like $f(g(x))$ written out." Truth: It applies whenever the argument of a function is anything other than a bare $x$ , including $e^{2x}$ , $\ln 5x$ , and $\sqrt{x+1}$ .
Myth 2: "The inner derivative is added to the outside derivative." Truth: The two pieces are multiplied, so a minus sign in the outside derivative survives and an addition never appears.
Myth 3: "A constant inside needs no factor." Truth: A constant inside contributes a constant factor, so $\sin 3x$ picks up a $3$ and $\sin(x + 1)$ picks up a $1$ that is invisible but present.

Topic-Specific Study Method

Name the Inside, Then Differentiate

Exam disguise	What you see on the page	What your brain must translate	Required execution
A linear inside	$\sin 5x$ , $e^{3x}$ , $\sqrt{2x+1}$	The argument carries a constant slope	Multiply by that constant, even when it is $1$
A power inside	$\sin(x^2)$ , $e^{x^2}$	The inside is not linear	Multiply by $2x$ , not by $2$
A function inside	$\sin(\cos x)$ , $e^{\sin x}$	Two layers, so two factors	Differentiate outward, multiplying at each layer
A power of a function	$\sin^2 x$ , $(x^2+1)^3$	The exponent is outside, the function is inside	Bring the power down, keep the inside, then multiply

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\frac{d}{dx}\left[\sin(3x) + e^{x^2}\right]$ .

Standard Execution:

Split. The sum rule lets each term be handled alone.

First term. The outside is sine, the inside is $3x$ , so the derivative is $3\cos 3x$ .

Second term. The outside is the exponential, the inside is $x^2$ , so the derivative is $2x\,e^{x^2}$ .

Assemble. The derivative is $3\cos 3x + 2x\,e^{x^2}$ .

Stop. No simplification is possible, and none is required.

Exam Variation (Context / Parameter Shift): Let $f(x) = (2x+1)^5$ . Find the equation of the line tangent to the graph of $f$ at $x = 1$ .

Solution: The chain rule gives $f'(x) = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4$ . At $x = 1$ the point is $(1, 243)$ and the slope is $f'(1) = 10 \cdot 81 = 810$ . The tangent line is $y - 243 = 810(x - 1)$ . The factor of $2$ inside the power is the entire difference between $810$ and the $405$ that a missing chain rule would produce.

Common Traps & Failure Signals:

The single factor (Signal: an answer with one multiplicative piece): the inside was differentiated mentally and then dropped instead of written.
The wrong evaluation point (Signal: $\cos x$ where the page had $\cos 3x$ ): the outside was differentiated at $x$ rather than at the untouched inside.
The early cancellation (Signal: a factor that disappears between two lines): simplification ran ahead of the second factor, and the algebra quietly absorbed it.

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

Inner Drop $\rightarrow$ Write the second factor first.
Layer Confusion: You named the outside function as the inside one, so the first factor in your answer was evaluated at the wrong expression and the remaining factor had nothing left to multiply.
Flat Reading: You saw a single function where the page had two, so the answer came out with one factor instead of two.

Past 10-Year AP Exam Appearance

Testing Frequency: Every Single Year; Every released exam from 2017 to 2026 contains at least one scored part whose answer cannot be reached without differentiating a composite
10-Year Total Count: 9 of 9 Exams · 21 FRQ Parts; Twenty-one parts across the nine released sets, counted part by part; the chain is scored through the derivative line rather than through a named part
FRQ Years Appeared: 2017 (Q5a, Q6a, Q6b, Q6c), 2018 (Q2a), 2019 (Q1d, Q6b, Q6c), 2021 (Q2c), 2022 (Q1c, Q2b), 2023 (Q1d, Q2b, Q5a, Q5b, Q6d), 2024 (Q2b, Q5d), 2025 (Q5A, Q6D), 2026 (Q6B)

Practice Examples

MCQ 1

The Linear Inside

If $f(x) = \sin(3x)$ , what is $f'(x)$ ?

MCQ 2

A Power Around a Linear

Let $g(x) = (2x+1)^5$ . What is $g'(0)$ ?

MCQ 3

The Exponential Trap

What is the derivative of $h(x) = e^{x^2}$ ?

FRQ 4

A Radical and a Tangent

Let $f$ be the function defined by $f(x) = \sqrt{3x + 1}$ for $x \ge -\frac{1}{3}$ .

(a) Find $f'(x)$ .

(b) Find the equation of the line tangent to the graph of $f$ at $x = 1$ .

(c) The function $g$ is defined by $g(x) = f(x^2)$ . Find $g'(x)$ and evaluate $g'(1)$ .

(d) A classmate writes $f'(x) = \frac{1}{2\sqrt{3x+1}}$ and claims the answer is complete. Identify the missing piece and explain what it changes about the slope at $x = 1$ .

Cheatsheet

The Chain Rule

	Move
Argument is a bare $x$	No chain rule needed
Argument is $ax + b$	Multiply by $a$
Argument is a function of $x$	Multiply by its derivative
Two layers	Two factors, one per layer
Three layers	Three factors, one per layer
Inside is a constant	The derivative is zero, since the function is constant

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Unit 3The Chain Rule Basics

Concept 4.129Built for May 2027 AP Exam

The Chain Rule Basics

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A sine, an exponential, a logarithm, or a power whose argument is not a bare $x$ , together with a request for a derivative, a slope, or an instantaneous rate.
What you’ll do: Name the outside and the inside, differentiate the outside at the untouched inside, then multiply by the inside derivative.
Memory hook: Outside first, inside stays, times inside. The inner derivative is a factor you multiply by, so it can never turn into a plus sign.
Use it on the exam: For MCQs: Every distractor family is built from the same break. The option that omits the inner derivative is always present, so check the number of factors in your answer before you check the algebra. For FRQs: When a part asks for $\frac{d}{dx}\left[e^{f(x)}\right]$ or $\frac{d}{dt}\left[\sin(g(t))\right]$ , the justification line must exhibit the inner factor. An answer that only differentiates the outside earns the form point and loses the reasoning point.

Concept Review

Two Factors, One Motion

Figure 1 — The chain rule produces two factors, not one. The outside function is differentiated at the untouched inside, and the derivative of the inside arrives as a multiplicand.

OUTER DERIVATIVE TIMES INNER DERIVATIVE

Reading the Nest Before You Differentiate

The only decision the rule requires is which layer is outside. Everything after that is bookkeeping.

Function	Outside $f$	Inside $g$	Derivative
$\sin 3x$	$\sin u$	$3x$	$3\cos 3x$
$e^{x^2}$	$e^u$	$x^2$	$2x\,e^{x^2}$
$(2x+1)^5$	$u^5$	$2x+1$	$10(2x+1)^4$
$\ln(x^2+1)$	$\ln u$	$x^2+1$	$\dfrac{2x}{x^2+1}$
$\sqrt{4x-3}$	$u^{1/2}$	$4x-3$	$\dfrac{2}{\sqrt{4x-3}}$

The Four Moves, in Order

Name the inside. Write $g(x) =$ the expression sitting in the argument, and confirm the original really is $f(g(x))$ .
Differentiate the outside at the untouched inside. For $\sin 3x$ that is $\cos 3x$ , not $\cos x$ .
Multiply by the inside derivative. This factor is what the exam is testing, and it is the one most often absent.
Simplify only afterwards. Cancelling before the second factor is written is how the factor gets lost.

Common Misconception Check

Myth 1: "The chain rule applies only when the function looks like $f(g(x))$ written out." Truth: It applies whenever the argument of a function is anything other than a bare $x$ , including $e^{2x}$ , $\ln 5x$ , and $\sqrt{x+1}$ .
Myth 2: "The inner derivative is added to the outside derivative." Truth: The two pieces are multiplied, so a minus sign in the outside derivative survives and an addition never appears.
Myth 3: "A constant inside needs no factor." Truth: A constant inside contributes a constant factor, so $\sin 3x$ picks up a $3$ and $\sin(x + 1)$ picks up a $1$ that is invisible but present.

Topic-Specific Study Method

Name the Inside, Then Differentiate

Exam disguise	What you see on the page	What your brain must translate	Required execution
A linear inside	$\sin 5x$ , $e^{3x}$ , $\sqrt{2x+1}$	The argument carries a constant slope	Multiply by that constant, even when it is $1$
A power inside	$\sin(x^2)$ , $e^{x^2}$	The inside is not linear	Multiply by $2x$ , not by $2$
A function inside	$\sin(\cos x)$ , $e^{\sin x}$	Two layers, so two factors	Differentiate outward, multiplying at each layer
A power of a function	$\sin^2 x$ , $(x^2+1)^3$	The exponent is outside, the function is inside	Bring the power down, keep the inside, then multiply

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\frac{d}{dx}\left[\sin(3x) + e^{x^2}\right]$ .

Standard Execution:

Split. The sum rule lets each term be handled alone.

First term. The outside is sine, the inside is $3x$ , so the derivative is $3\cos 3x$ .

Second term. The outside is the exponential, the inside is $x^2$ , so the derivative is $2x\,e^{x^2}$ .

Assemble. The derivative is $3\cos 3x + 2x\,e^{x^2}$ .

Stop. No simplification is possible, and none is required.

Exam Variation (Context / Parameter Shift): Let $f(x) = (2x+1)^5$ . Find the equation of the line tangent to the graph of $f$ at $x = 1$ .

Solution: The chain rule gives $f'(x) = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4$ . At $x = 1$ the point is $(1, 243)$ and the slope is $f'(1) = 10 \cdot 81 = 810$ . The tangent line is $y - 243 = 810(x - 1)$ . The factor of $2$ inside the power is the entire difference between $810$ and the $405$ that a missing chain rule would produce.

Common Traps & Failure Signals:

The single factor (Signal: an answer with one multiplicative piece): the inside was differentiated mentally and then dropped instead of written.
The wrong evaluation point (Signal: $\cos x$ where the page had $\cos 3x$ ): the outside was differentiated at $x$ rather than at the untouched inside.
The early cancellation (Signal: a factor that disappears between two lines): simplification ran ahead of the second factor, and the algebra quietly absorbed it.

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

Inner Drop $\rightarrow$ Write the second factor first.
Layer Confusion: You named the outside function as the inside one, so the first factor in your answer was evaluated at the wrong expression and the remaining factor had nothing left to multiply.
Flat Reading: You saw a single function where the page had two, so the answer came out with one factor instead of two.

Past 10-Year AP Exam Appearance

Testing Frequency: Every Single Year; Every released exam from 2017 to 2026 contains at least one scored part whose answer cannot be reached without differentiating a composite
10-Year Total Count: 9 of 9 Exams · 21 FRQ Parts; Twenty-one parts across the nine released sets, counted part by part; the chain is scored through the derivative line rather than through a named part
FRQ Years Appeared: 2017 (Q5a, Q6a, Q6b, Q6c), 2018 (Q2a), 2019 (Q1d, Q6b, Q6c), 2021 (Q2c), 2022 (Q1c, Q2b), 2023 (Q1d, Q2b, Q5a, Q5b, Q6d), 2024 (Q2b, Q5d), 2025 (Q5A, Q6D), 2026 (Q6B)

Practice Examples

MCQ 1

The Linear Inside

If $f(x) = \sin(3x)$ , what is $f'(x)$ ?

MCQ 2

A Power Around a Linear

Let $g(x) = (2x+1)^5$ . What is $g'(0)$ ?

MCQ 3

The Exponential Trap

What is the derivative of $h(x) = e^{x^2}$ ?

FRQ 4

A Radical and a Tangent

Let $f$ be the function defined by $f(x) = \sqrt{3x + 1}$ for $x \ge -\frac{1}{3}$ .

(a) Find $f'(x)$ .

(b) Find the equation of the line tangent to the graph of $f$ at $x = 1$ .

(c) The function $g$ is defined by $g(x) = f(x^2)$ . Find $g'(x)$ and evaluate $g'(1)$ .

(d) A classmate writes $f'(x) = \frac{1}{2\sqrt{3x+1}}$ and claims the answer is complete. Identify the missing piece and explain what it changes about the slope at $x = 1$ .

Cheatsheet

The Chain Rule

	Move
Argument is a bare $x$	No chain rule needed
Argument is $ax + b$	Multiply by $a$
Argument is a function of $x$	Multiply by its derivative
Two layers	Two factors, one per layer
Three layers	Three factors, one per layer
Inside is a constant	The derivative is zero, since the function is constant

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