Unit 2The Power Rule

Concept 4.135Built for May 2027 AP Exam

The Power Rule

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The power rule replaces the definition of the derivative for every term of the form $x^n$ . Bring the exponent down as a multiplier, then subtract one from it: $\frac{d}{dx}x^n = nx^{n-1}$ . The rule is not restricted to positive whole numbers. A radical is a fractional exponent, a reciprocal is a negative exponent, and a constant term is the case $n = 0$ , which correctly produces zero. What the rule does not touch is the coefficient, which travels along unchanged. Nearly every error in this topic happens before the rule is applied, when a square root or a fraction is left in its original shape and the exponent never gets isolated. Rewrite first, differentiate second.

On the AP Exam

This rule is a tool rather than a topic. It is scored inside larger questions, usually as the first step of a tangent line, a rate of change, or a critical point, and a slip here contaminates every later part of the answer. The grader awards the derivative expression itself, so the points for this skill are collected or lost in a single line of work.

What you’ll see: A polynomial, a sum involving radicals, a rational term, or a request for a tangent line or a rate of change where the function is given by a formula.
What you’ll do: Convert every term to power form, differentiate term by term, and simplify only if a later part needs it.
Memory hook: Down, then one less. The exponent becomes the multiplier first, and only then does the exponent itself shrink.
Use it on the exam: For MCQs: Watch for a term such as $\frac{4}{x}$ , which is $4x^{-1}$ and differentiates to $-4x^{-2}$ , not to a logarithm and not to zero. For FRQs: When the question says "it can be shown that", the given derivative usually came from this rule. Check the given line against your own work rather than trusting it blindly, then use the verified version.

Concept Review

Figure 1

One motion, three disguises. The rule never touches the coefficient, and the only work it demands is the rewriting that turns a radical or a reciprocal into a bare power of $x$ .

THE RULE IS ONE MOTION, REPEATED

The Rule and Its Three Rewrites

Written in the problem	Power form	Derivative
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\frac{1}{x^3}$	$x^{-3}$	$-3x^{-4}$
$\frac{1}{\sqrt[3]{x}}$	$x^{-1/3}$	$-\frac{1}{3}x^{-4/3}$
$7$	$7x^{0}$	$0$

The exponent is the only thing that changes. A coefficient in front of the power stays in front, so $5x^4$ becomes $20x^3$ and $\frac{1}{4}x^8$ becomes $2x^7$ . A constant term follows the same rule with $n = 0$ , which gives $0 \cdot x^{-1} = 0$ ; that is why constants vanish rather than becoming $x^{-1}$ .

Where the Rule Comes From

For a positive integer $n$ , expand the difference quotient:

$\frac{(x+h)^n - x^n}{h} = \frac{x^n + n x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots - x^n}{h} = n x^{n-1} + \binom{n}{2}x^{n-2}h + \cdots$

Every term after the first carries a factor of $h$ , so the limit as $h \to 0$ leaves only $n x^{n-1}$ . The argument explains the shape of the result rather than just the result: exactly one term survives the division, the one whose $h$ cancels, and its coefficient is the number of ways to pick a single $h$ from $n$ factors.

When the Rule Does Not Apply

Expression	Rule to use	Why
$3^x$	Exponential rule	The variable sits in the exponent, not the base
$x^x$	Neither rule alone	Both base and exponent move
$\sin^2 x$	Chain rule	The power sits outside a function of $x$
$\lvert x \rvert$	None at the origin	There is no exponent to bring down

The test is mechanical: the rule applies when the base is a bare $x$ and the exponent is a constant. If either condition fails, the expression needs a different tool, and the power rule applied anyway will produce a wrong answer that looks plausible.

Common Misconception Check

Myth 1: "A constant differentiates to the constant itself." Truth: A constant has slope zero everywhere, since its graph is horizontal. The value of the constant is irrelevant to its derivative.
Myth 2: "The exponent becomes $n$ in the new exponent as well." Truth: The exponent is used twice, once as the multiplier and once decremented. Writing $n x^n$ is the single most common arithmetic slip in this topic.
Myth 3: "The rule works for $3^x$ because it is written as a power." Truth: The rule requires a constant exponent. For $3^x$ the exponent is the variable, so the answer is $3^x \ln 3$ , not $x3^{x-1}$ .

Topic-Specific Study Method

Rewrite, Then Differentiate

Borrowed idea: The power rule is a single motion repeated: bring the exponent down in front, then drop that exponent by one. Nothing else moves. A coefficient travels along untouched, because scaling a function scales its slope by the same factor. The rule survives every disguise the exam can apply, since a radical is a fractional exponent, a reciprocal is a negative exponent, and a lone constant is the case $n = 0$ . Almost every lost point comes from a rewriting mistake made before the rule was applied, not from the rule itself, which means the skill actually being tested is converting the expression into power form before the differentiating begins.

Exam disguise	What you see on the page	What your brain must translate	Required execution
A radical	$\sqrt{x}$ or $\sqrt[4]{x^3}$	Fractional exponents	Write $x^{1/2}$ or $x^{3/4}$ first
A fraction with $x$ below	$\frac{5}{x^2}$	A negative exponent with a coefficient	Rewrite as $5x^{-2}$
A constant	$9$	The case $n = 0$	Differentiate to $0$ , not to $9$
A tangent-line request	"the slope of the line tangent to $f$ at $x = 4$ "	A derivative evaluated at one input	Differentiate, then substitute

① Take $f(x) = 3\sqrt{x} - \frac{2}{x} + 5$ and write it as $3x^{1/2} - 2x^{-1} + 5x^0$ before differentiating. Compare the two versions on the page and notice that the second one has no decision left to make.

② Build a table of five expressions your teacher has used that are not in power form, and convert each one. The conversion is the whole skill, and it takes about a minute to practise.

③ On any tangent-line question, differentiate first and substitute second, never the other way around. Substituting early produces a number with no slope attached to it.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\frac{d}{dx}\left(4x^3 - \frac{6}{x^2} + \sqrt{x}\right)$ .

Standard Execution:

Convert. The expression becomes $4x^3 - 6x^{-2} + x^{1/2}$ .

Differentiate each term. $12x^2$ , then $12x^{-3}$ , then $\frac{1}{2}x^{-1/2}$ .

Watch the signs. Subtracting $6x^{-2}$ produces $+12x^{-3}$ , because the exponent is negative.

Assemble. The derivative is $12x^2 + 12x^{-3} + \frac{1}{2}x^{-1/2}$ .

Stop. Simplification is optional unless a later part asks for a value.

Exam Variation (Context / Parameter Shift): The curve $y = x^3 - 6x^2 + 9x$ has two points where the tangent is horizontal. Find them.

Solution: Differentiate to get $y' = 3x^2 - 12x + 9 = 3(x-1)(x-3)$ . A horizontal tangent requires $y' = 0$ , so $x = 1$ or $x = 3$ . Substituting gives the points $(1, 4)$ and $(3, 0)$ . The derivative answers the question and the original function supplies the heights, which is why both must be kept on the page.

Common Traps & Failure Signals:

The untouched radical (Signal: a square root surviving into the answer): the expression was never rewritten, so the rule had no exponent to work with.
The single-step exponent (Signal: an answer with the original exponent intact): the exponent was used as a multiplier but never decremented.
The vanishing coefficient (Signal: a coefficient of one where the problem had three): the multiplier was dropped during the rewrite.

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

Exponent Drift $\rightarrow$ Circle the new exponent before writing the term.
Coefficient Loss: You left the constant behind.
Radical Freeze: You stopped at the square root and never rewrote it as a fractional power, so the rule had no exponent to bring down and the entire term was quietly abandoned instead of being differentiated.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; The rule is scored inside larger parts rather than as a standalone item, and it carries the first step of tangent-line, rate-of-change, and critical-point work
10-Year Total Count: 6 of 9 Exams · 7 FRQ Parts
FRQ Years Appeared: 2017 (Q5c), 2018 (Q2a, Q2d), 2019 (Q1d), 2021 (Q2c), 2023 (Q5b), 2025 (Q2D)

Practice Examples

MCQ 1

Radical in Disguise

If $f(x) = 6\sqrt{x}$ , what is $f'(9)$ ?

MCQ 2

Negative Exponent

What is the derivative of $g(x) = \frac{8}{x^2}$ ?

MCQ 3

The Constant Case

Let $h(x) = 5x^4 - 12$ . What is $h'(x)$ ?

FRQ 4

Tangents on a Cubic

Let $f$ be the function defined by $f(x) = 2x^3 - 9x^2 + 12x - 4$ for all real $x$ .

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

(b) Find the $x$ -coordinates of all points on the graph of $f$ where the tangent line is horizontal.

(c) Write an equation for the line tangent to the graph of $f$ at $x = 1$ .

(d) The function $g$ is defined by $g(x) = \frac{f(x)}{x}$ for $x \ne 0$ . Find $g'(x)$ and evaluate $g'(2)$ .

Cheatsheet

The Power Rule

	Move
$x^n$ with $n$ constant	$nx^{n-1}$
Coefficient in front	Multiply it, do not change it
Radical	Rewrite as a fractional exponent
Variable in a denominator	Rewrite as a negative exponent
Constant alone	Derivative is $0$
Variable in the exponent	Stop: this needs the exponential rule

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Unit 2The Power Rule

Concept 4.135Built for May 2027 AP Exam

The Power Rule

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 involving radicals, a rational term, or a request for a tangent line or a rate of change where the function is given by a formula.
What you’ll do: Convert every term to power form, differentiate term by term, and simplify only if a later part needs it.
Memory hook: Down, then one less. The exponent becomes the multiplier first, and only then does the exponent itself shrink.
Use it on the exam: For MCQs: Watch for a term such as $\frac{4}{x}$ , which is $4x^{-1}$ and differentiates to $-4x^{-2}$ , not to a logarithm and not to zero. For FRQs: When the question says "it can be shown that", the given derivative usually came from this rule. Check the given line against your own work rather than trusting it blindly, then use the verified version.

Concept Review

Figure 1

One motion, three disguises. The rule never touches the coefficient, and the only work it demands is the rewriting that turns a radical or a reciprocal into a bare power of $x$ .

THE RULE IS ONE MOTION, REPEATED

The Rule and Its Three Rewrites

Written in the problem	Power form	Derivative
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\frac{1}{x^3}$	$x^{-3}$	$-3x^{-4}$
$\frac{1}{\sqrt[3]{x}}$	$x^{-1/3}$	$-\frac{1}{3}x^{-4/3}$
$7$	$7x^{0}$	$0$

Where the Rule Comes From

For a positive integer $n$ , expand the difference quotient:

$\frac{(x+h)^n - x^n}{h} = \frac{x^n + n x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots - x^n}{h} = n x^{n-1} + \binom{n}{2}x^{n-2}h + \cdots$

When the Rule Does Not Apply

Expression	Rule to use	Why
$3^x$	Exponential rule	The variable sits in the exponent, not the base
$x^x$	Neither rule alone	Both base and exponent move
$\sin^2 x$	Chain rule	The power sits outside a function of $x$
$\lvert x \rvert$	None at the origin	There is no exponent to bring down

Common Misconception Check

Myth 1: "A constant differentiates to the constant itself." Truth: A constant has slope zero everywhere, since its graph is horizontal. The value of the constant is irrelevant to its derivative.
Myth 2: "The exponent becomes $n$ in the new exponent as well." Truth: The exponent is used twice, once as the multiplier and once decremented. Writing $n x^n$ is the single most common arithmetic slip in this topic.
Myth 3: "The rule works for $3^x$ because it is written as a power." Truth: The rule requires a constant exponent. For $3^x$ the exponent is the variable, so the answer is $3^x \ln 3$ , not $x3^{x-1}$ .

Topic-Specific Study Method

Rewrite, Then Differentiate

Exam disguise	What you see on the page	What your brain must translate	Required execution
A radical	$\sqrt{x}$ or $\sqrt[4]{x^3}$	Fractional exponents	Write $x^{1/2}$ or $x^{3/4}$ first
A fraction with $x$ below	$\frac{5}{x^2}$	A negative exponent with a coefficient	Rewrite as $5x^{-2}$
A constant	$9$	The case $n = 0$	Differentiate to $0$ , not to $9$
A tangent-line request	"the slope of the line tangent to $f$ at $x = 4$ "	A derivative evaluated at one input	Differentiate, then substitute

② Build a table of five expressions your teacher has used that are not in power form, and convert each one. The conversion is the whole skill, and it takes about a minute to practise.

③ On any tangent-line question, differentiate first and substitute second, never the other way around. Substituting early produces a number with no slope attached to it.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\frac{d}{dx}\left(4x^3 - \frac{6}{x^2} + \sqrt{x}\right)$ .

Standard Execution:

Convert. The expression becomes $4x^3 - 6x^{-2} + x^{1/2}$ .

Differentiate each term. $12x^2$ , then $12x^{-3}$ , then $\frac{1}{2}x^{-1/2}$ .

Watch the signs. Subtracting $6x^{-2}$ produces $+12x^{-3}$ , because the exponent is negative.

Assemble. The derivative is $12x^2 + 12x^{-3} + \frac{1}{2}x^{-1/2}$ .

Stop. Simplification is optional unless a later part asks for a value.

Exam Variation (Context / Parameter Shift): The curve $y = x^3 - 6x^2 + 9x$ has two points where the tangent is horizontal. Find them.

Solution: Differentiate to get $y' = 3x^2 - 12x + 9 = 3(x-1)(x-3)$ . A horizontal tangent requires $y' = 0$ , so $x = 1$ or $x = 3$ . Substituting gives the points $(1, 4)$ and $(3, 0)$ . The derivative answers the question and the original function supplies the heights, which is why both must be kept on the page.

Common Traps & Failure Signals:

The untouched radical (Signal: a square root surviving into the answer): the expression was never rewritten, so the rule had no exponent to work with.
The single-step exponent (Signal: an answer with the original exponent intact): the exponent was used as a multiplier but never decremented.
The vanishing coefficient (Signal: a coefficient of one where the problem had three): the multiplier was dropped during the rewrite.

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

Exponent Drift $\rightarrow$ Circle the new exponent before writing the term.
Coefficient Loss: You left the constant behind.
Radical Freeze: You stopped at the square root and never rewrote it as a fractional power, so the rule had no exponent to bring down and the entire term was quietly abandoned instead of being differentiated.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; The rule is scored inside larger parts rather than as a standalone item, and it carries the first step of tangent-line, rate-of-change, and critical-point work
10-Year Total Count: 6 of 9 Exams · 7 FRQ Parts
FRQ Years Appeared: 2017 (Q5c), 2018 (Q2a, Q2d), 2019 (Q1d), 2021 (Q2c), 2023 (Q5b), 2025 (Q2D)

Practice Examples

MCQ 1

Radical in Disguise

If $f(x) = 6\sqrt{x}$ , what is $f'(9)$ ?

MCQ 2

Negative Exponent

What is the derivative of $g(x) = \frac{8}{x^2}$ ?

MCQ 3

The Constant Case

Let $h(x) = 5x^4 - 12$ . What is $h'(x)$ ?

FRQ 4

Tangents on a Cubic

Let $f$ be the function defined by $f(x) = 2x^3 - 9x^2 + 12x - 4$ for all real $x$ .

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

(b) Find the $x$ -coordinates of all points on the graph of $f$ where the tangent line is horizontal.

(c) Write an equation for the line tangent to the graph of $f$ at $x = 1$ .

(d) The function $g$ is defined by $g(x) = \frac{f(x)}{x}$ for $x \ne 0$ . Find $g'(x)$ and evaluate $g'(2)$ .

Cheatsheet

The Power Rule

	Move
$x^n$ with $n$ constant	$nx^{n-1}$
Coefficient in front	Multiply it, do not change it
Radical	Rewrite as a fractional exponent
Variable in a denominator	Rewrite as a negative exponent
Constant alone	Derivative is $0$
Variable in the exponent	Stop: this needs the exponential rule

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Start Topic Practice Next Topic: 2.8 Constant Multiple and Sum/Difference Rules

Unit 2The Power Rule

Concept 4.135Built for May 2027 AP Exam

The Power Rule

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 involving radicals, a rational term, or a request for a tangent line or a rate of change where the function is given by a formula.
What you’ll do: Convert every term to power form, differentiate term by term, and simplify only if a later part needs it.
Memory hook: Down, then one less. The exponent becomes the multiplier first, and only then does the exponent itself shrink.
Use it on the exam: For MCQs: Watch for a term such as $\frac{4}{x}$ , which is $4x^{-1}$ and differentiates to $-4x^{-2}$ , not to a logarithm and not to zero. For FRQs: When the question says "it can be shown that", the given derivative usually came from this rule. Check the given line against your own work rather than trusting it blindly, then use the verified version.

Concept Review

Figure 1

One motion, three disguises. The rule never touches the coefficient, and the only work it demands is the rewriting that turns a radical or a reciprocal into a bare power of $x$ .

THE RULE IS ONE MOTION, REPEATED

The Rule and Its Three Rewrites

Written in the problem	Power form	Derivative
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\frac{1}{x^3}$	$x^{-3}$	$-3x^{-4}$
$\frac{1}{\sqrt[3]{x}}$	$x^{-1/3}$	$-\frac{1}{3}x^{-4/3}$
$7$	$7x^{0}$	$0$

Where the Rule Comes From

For a positive integer $n$ , expand the difference quotient:

$\frac{(x+h)^n - x^n}{h} = \frac{x^n + n x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots - x^n}{h} = n x^{n-1} + \binom{n}{2}x^{n-2}h + \cdots$

When the Rule Does Not Apply

Expression	Rule to use	Why
$3^x$	Exponential rule	The variable sits in the exponent, not the base
$x^x$	Neither rule alone	Both base and exponent move
$\sin^2 x$	Chain rule	The power sits outside a function of $x$
$\lvert x \rvert$	None at the origin	There is no exponent to bring down

Common Misconception Check

Myth 1: "A constant differentiates to the constant itself." Truth: A constant has slope zero everywhere, since its graph is horizontal. The value of the constant is irrelevant to its derivative.
Myth 2: "The exponent becomes $n$ in the new exponent as well." Truth: The exponent is used twice, once as the multiplier and once decremented. Writing $n x^n$ is the single most common arithmetic slip in this topic.
Myth 3: "The rule works for $3^x$ because it is written as a power." Truth: The rule requires a constant exponent. For $3^x$ the exponent is the variable, so the answer is $3^x \ln 3$ , not $x3^{x-1}$ .

Topic-Specific Study Method

Rewrite, Then Differentiate

Exam disguise	What you see on the page	What your brain must translate	Required execution
A radical	$\sqrt{x}$ or $\sqrt[4]{x^3}$	Fractional exponents	Write $x^{1/2}$ or $x^{3/4}$ first
A fraction with $x$ below	$\frac{5}{x^2}$	A negative exponent with a coefficient	Rewrite as $5x^{-2}$
A constant	$9$	The case $n = 0$	Differentiate to $0$ , not to $9$
A tangent-line request	"the slope of the line tangent to $f$ at $x = 4$ "	A derivative evaluated at one input	Differentiate, then substitute

② Build a table of five expressions your teacher has used that are not in power form, and convert each one. The conversion is the whole skill, and it takes about a minute to practise.

③ On any tangent-line question, differentiate first and substitute second, never the other way around. Substituting early produces a number with no slope attached to it.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\frac{d}{dx}\left(4x^3 - \frac{6}{x^2} + \sqrt{x}\right)$ .

Standard Execution:

Convert. The expression becomes $4x^3 - 6x^{-2} + x^{1/2}$ .

Differentiate each term. $12x^2$ , then $12x^{-3}$ , then $\frac{1}{2}x^{-1/2}$ .

Watch the signs. Subtracting $6x^{-2}$ produces $+12x^{-3}$ , because the exponent is negative.

Assemble. The derivative is $12x^2 + 12x^{-3} + \frac{1}{2}x^{-1/2}$ .

Stop. Simplification is optional unless a later part asks for a value.

Exam Variation (Context / Parameter Shift): The curve $y = x^3 - 6x^2 + 9x$ has two points where the tangent is horizontal. Find them.

Solution: Differentiate to get $y' = 3x^2 - 12x + 9 = 3(x-1)(x-3)$ . A horizontal tangent requires $y' = 0$ , so $x = 1$ or $x = 3$ . Substituting gives the points $(1, 4)$ and $(3, 0)$ . The derivative answers the question and the original function supplies the heights, which is why both must be kept on the page.

Common Traps & Failure Signals:

The untouched radical (Signal: a square root surviving into the answer): the expression was never rewritten, so the rule had no exponent to work with.
The single-step exponent (Signal: an answer with the original exponent intact): the exponent was used as a multiplier but never decremented.
The vanishing coefficient (Signal: a coefficient of one where the problem had three): the multiplier was dropped during the rewrite.

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

Exponent Drift $\rightarrow$ Circle the new exponent before writing the term.
Coefficient Loss: You left the constant behind.
Radical Freeze: You stopped at the square root and never rewrote it as a fractional power, so the rule had no exponent to bring down and the entire term was quietly abandoned instead of being differentiated.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; The rule is scored inside larger parts rather than as a standalone item, and it carries the first step of tangent-line, rate-of-change, and critical-point work
10-Year Total Count: 6 of 9 Exams · 7 FRQ Parts
FRQ Years Appeared: 2017 (Q5c), 2018 (Q2a, Q2d), 2019 (Q1d), 2021 (Q2c), 2023 (Q5b), 2025 (Q2D)

Practice Examples

MCQ 1

Radical in Disguise

If $f(x) = 6\sqrt{x}$ , what is $f'(9)$ ?

MCQ 2

Negative Exponent

What is the derivative of $g(x) = \frac{8}{x^2}$ ?

MCQ 3

The Constant Case

Let $h(x) = 5x^4 - 12$ . What is $h'(x)$ ?

FRQ 4

Tangents on a Cubic

Let $f$ be the function defined by $f(x) = 2x^3 - 9x^2 + 12x - 4$ for all real $x$ .

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

(b) Find the $x$ -coordinates of all points on the graph of $f$ where the tangent line is horizontal.

(c) Write an equation for the line tangent to the graph of $f$ at $x = 1$ .

(d) The function $g$ is defined by $g(x) = \frac{f(x)}{x}$ for $x \ne 0$ . Find $g'(x)$ and evaluate $g'(2)$ .

Cheatsheet

The Power Rule

	Move
$x^n$ with $n$ constant	$nx^{n-1}$
Coefficient in front	Multiply it, do not change it
Radical	Rewrite as a fractional exponent
Variable in a denominator	Rewrite as a negative exponent
Constant alone	Derivative is $0$
Variable in the exponent	Stop: this needs the exponential rule

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