Unit 2Derivative of

x^n

Concept 4.224Built for May 2027 AP Exam

Derivative of $x^n$

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The power rule is stated for a constant exponent, and "constant" is broader than "whole number". $\frac{d}{dx}x^n = nx^{n-1}$ holds for every real $n$ , which means roots and reciprocals are already covered once they are rewritten as powers: $\sqrt{x} = x^{1/2}$ and $\frac{1}{x^3} = x^{-3}$ . The derivation does not assume $n$ is an integer either. Expand $(x+h)^n$ , and every term after the first carries a factor of $h$ , so the limit leaves exactly $nx^{n-1}$ . Two consequences matter on the exam. The rewrite is the work, not the differentiation. And a fractional power can fail at the endpoint of its domain, where the slope becomes infinite rather than finite.

On the AP Exam

What you’ll see: A term with a radical, a reciprocal, or a power of a power, inside a rate, a tangent slope, or a second derivative.
What you’ll do: Convert to $x^n$ , differentiate with one step, then simplify back into the form the answer choices or the rubric expect.
Memory hook: A root is a fraction, a reciprocal is a negative. Costume first, rule second.
Use it on the exam: For MCQs: Options are frequently the same expression written two ways, one with a radical and one with a fractional exponent. Simplify back to the original costume before you compare. For FRQs: The rubric accepts a derivative in either form, but a later part usually substitutes a number. A negative exponent is easier to evaluate than a compound fraction, so convert toward whichever is easier to plug into.

Concept Review

Same Rule, Different Costumes

Figure 1 — Each row is one term in three states. The middle column is the only real work: once the term is written as a power, the derivative follows in a single step and the rule itself never changes.

Three rows mapping a radical or reciprocal in its original form to a fractional or negative power and then to its derivative, showing that the power rule is unchanged once the rewrite is done

The Rewrites Worth Memorising

Written as	As a power	Derivative	Derivative, original form
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$	$\frac{1}{2\sqrt{x}}$
$\sqrt[3]{x}$	$x^{1/3}$	$\frac{1}{3}x^{-2/3}$	$\frac{1}{3\sqrt[3]{x^2}}$
$\frac{1}{x}$	$x^{-1}$	$-x^{-2}$	$-\frac{1}{x^2}$
$\frac{1}{x^4}$	$x^{-4}$	$-4x^{-5}$	$-\frac{4}{x^5}$
$x\sqrt{x}$	$x^{3/2}$	$\frac{3}{2}x^{1/2}$	$\frac{3\sqrt{x}}{2}$

The right-hand column is what the answer key usually prints, and the two middle columns are what you actually compute. Translating between them is a skill in itself, and it is the one that decides whether an otherwise correct derivative matches an option.

Where the Rule Comes From

Expand $(x+h)^n$ with the binomial theorem and subtract $x^n$ :

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

Divide by $h$ and every term after the first still carries a factor of $h$ :

$\lim_{h \to 0}\frac{(x+h)^n - x^n}{h} = nx^{n-1} + 0 + \cdots = nx^{n-1}$

The exponent drops because exactly one of the $n$ factors is replaced by the increment, and there are $n$ ways to choose it. Nothing in that counting argument requires $n$ to be a whole number, which is why the fractional case inherits the rule rather than needing a separate proof.

Where a Fractional Power Breaks

For $f(x) = \sqrt{x}$ the derivative $\frac{1}{2\sqrt{x}}$ is undefined at $x = 0$ , and the graph agrees: the tangent there is vertical, so no finite slope exists. This is the vertical-tangent failure from the previous topic, appearing inside a formula instead of a picture. The lesson is that the rule reports a slope wherever the rewritten power is differentiable, and the endpoint of a root's domain has to be checked separately.

Common Misconception Check

Myth 1: "The power rule only applies to positive integer exponents." Truth: It applies to every constant exponent. The integer case is the easiest, not the only one.
Myth 2: "The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$ by a separate rule." Truth: It is the same rule after one rewrite. Memorising it separately doubles the material for no gain.
Myth 3: "A negative exponent can be left in the denominator." Truth: It can, but the minus sign must be tracked. Most lost points in this topic come from dropping it during the translation back.

Topic-Specific Study Method

Rewrite, Then Drop

Borrowed idea: Rewrite before you differentiate. Every root and every reciprocal is a power wearing a costume. Turn the root into a fractional exponent, turn the reciprocal into a negative one, and the rule runs unchanged. Does the exponent have to be a whole number? No, the same algebra works for any constant, which is why the rewrite is worth the ten seconds. What about the endpoint where a root starts? There the slope may be infinite, and the derivative simply does not exist. So check where the expression lives before you report a number.

You see	Rewrite	Derivative	Watch for
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$	slope infinite at $0$
$\frac{1}{\sqrt{x}}$	$x^{-1/2}$	$-\frac{1}{2}x^{-3/2}$	sign and domain
$\frac{5}{x^2}$	$5x^{-2}$	$-10x^{-3}$	coefficient survives
$x^2\sqrt{x}$	$x^{5/2}$	$\frac{5}{2}x^{3/2}$	add the exponents first
$\frac{x^3}{x}$	$x^2$	$2x$	simplify before differentiating

① Write ten expressions in mixed costumes, convert each to a single power, and differentiate. Then convert every answer back to the original costume and check that the two forms agree at a test value.

② Differentiate $\sqrt{x}$ three ways: by the rule, by the definition with a conjugate, and by reading the graph at $x = 4$ . All three should give $\frac{1}{4}$ .

③ Find where $f(x) = x^{2/3}$ has a finite derivative and where it does not. The formula gives $\frac{2}{3}x^{-1/3}$ , which is undefined at $0$ , and that single point is the cusp from the previous topic.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = 3\sqrt{x} - \frac{2}{x}$ .

Standard Execution:

Rewrite. $f(x) = 3x^{1/2} - 2x^{-1}$ .

Differentiate. $f'(x) = \frac{3}{2}x^{-1/2} + 2x^{-2}$ .

Translate back. $f'(x) = \frac{3}{2\sqrt{x}} + \frac{2}{x^2}$ .

Check. At $x = 1$ both forms give $3.5$ , and the original function is decreasing in its reciprocal term, which matches the positive second term.

Exam Variation (Context / Parameter Shift): Water drains from a tank so that the depth at time $t$ minutes is $D(t) = 12\sqrt{9 - t}$ centimeters for $0 \le t < 9$ . Find the rate at which the depth is changing at $t = 5$ , with units, and explain what happens as $t \to 9^-$ .

Solution: Rewrite as $D(t) = 12(9-t)^{1/2}$ , so $D'(t) = 6(9-t)^{-1/2} \cdot (-1) = -\frac{6}{\sqrt{9-t}}$ . At $t = 5$ this gives $-3$ centimeters per minute. As $t \to 9^-$ the denominator tends to zero, so the rate becomes unbounded: the tank empties with a vertical tangent, which is the endpoint failure again.

Common Traps & Failure Signals:

The index as exponent (Signal: $\sqrt{x}$ differentiated to $\frac{1}{2}x^{1}$ or similar): the rewrite used the wrong power.
The lost sign (Signal: a positive answer where the function is decreasing): the negative exponent was translated back without its minus.
The unexamined endpoint (Signal: a finite slope reported at the start of a root): the domain was never checked.

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

Root Unrewritten: You never turned the root into a power.
Sign Lost $\rightarrow$ Carry the minus sign through the translation back.
Endpoint Ignored: You reported a finite slope at an input where the rewritten power has a negative exponent over a vanishing quantity, which is the formula's way of saying the tangent is vertical and no derivative exists there.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Fractional and negative exponents usually arrive already rewritten, and no released part from 2017 to 2026 scores the rewrite as an object of its own
10-Year Total Count: 1 of 9 Exams · 2 FRQ Parts; Both parts are 2019, and in both the rewritten power is the only route to the slope
FRQ Years Appeared: 2019 (Q4b, Q5a)

Practice Examples

MCQ 1

A Fractional Power

If $f(x) = x^{3/2}$ , what is $f'(x)$ ?

MCQ 2

A Reciprocal Root

If $g(x) = \frac{1}{\sqrt[3]{x}}$ , what is $g'(x)$ ?

MCQ 3

Why It Still Works

Why does the power rule apply to $x^{1/2}$ even though the exponent is not a whole number?

FRQ 1 • A Tank That Empties

Water drains from a tank so that the depth of water at time $t$ minutes is modeled by $D(t) = 12\sqrt{9 - t}$ centimeters for $0 \le t < 9$ .

(a) Write $D(t)$ as a power of $(9 - t)$ and find $D'(t)$ .

(b) Find the rate at which the depth is changing at $t = 5$ minutes. Include units.

(c) Explain why $D'(9)$ does not exist, using the form of $D'(t)$ rather than the graph.

(d) A second tank drains so that its depth is $E(t) = 12(9-t)^{1/3}$ . Compare $E'(5)$ with $D'(5)$ and explain which tank is draining faster at that instant.

Cheatsheet

Derivative of $x^n$

	Rewrite	Derivative
$x^n$	none	$nx^{n-1}$
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\sqrt[n]{x}$	$x^{1/n}$	$\frac{1}{n}x^{1/n - 1}$
$\frac{1}{x^n}$	$x^{-n}$	$-nx^{-n-1}$
$x^a x^b$	$x^{a+b}$	$(a+b)x^{a+b-1}$
$\frac{x^a}{x^b}$	$x^{a-b}$	$(a-b)x^{a-b-1}$

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Unit 2Derivative of

x^n

Concept 4.224Built for May 2027 AP Exam

Derivative of $x^n$

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A term with a radical, a reciprocal, or a power of a power, inside a rate, a tangent slope, or a second derivative.
What you’ll do: Convert to $x^n$ , differentiate with one step, then simplify back into the form the answer choices or the rubric expect.
Memory hook: A root is a fraction, a reciprocal is a negative. Costume first, rule second.
Use it on the exam: For MCQs: Options are frequently the same expression written two ways, one with a radical and one with a fractional exponent. Simplify back to the original costume before you compare. For FRQs: The rubric accepts a derivative in either form, but a later part usually substitutes a number. A negative exponent is easier to evaluate than a compound fraction, so convert toward whichever is easier to plug into.

Concept Review

Same Rule, Different Costumes

Three rows mapping a radical or reciprocal in its original form to a fractional or negative power and then to its derivative, showing that the power rule is unchanged once the rewrite is done

The Rewrites Worth Memorising

Written as	As a power	Derivative	Derivative, original form
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$	$\frac{1}{2\sqrt{x}}$
$\sqrt[3]{x}$	$x^{1/3}$	$\frac{1}{3}x^{-2/3}$	$\frac{1}{3\sqrt[3]{x^2}}$
$\frac{1}{x}$	$x^{-1}$	$-x^{-2}$	$-\frac{1}{x^2}$
$\frac{1}{x^4}$	$x^{-4}$	$-4x^{-5}$	$-\frac{4}{x^5}$
$x\sqrt{x}$	$x^{3/2}$	$\frac{3}{2}x^{1/2}$	$\frac{3\sqrt{x}}{2}$

Where the Rule Comes From

Expand $(x+h)^n$ with the binomial theorem and subtract $x^n$ :

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

Divide by $h$ and every term after the first still carries a factor of $h$ :

$\lim_{h \to 0}\frac{(x+h)^n - x^n}{h} = nx^{n-1} + 0 + \cdots = nx^{n-1}$

Where a Fractional Power Breaks

Common Misconception Check

Myth 1: "The power rule only applies to positive integer exponents." Truth: It applies to every constant exponent. The integer case is the easiest, not the only one.
Myth 2: "The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$ by a separate rule." Truth: It is the same rule after one rewrite. Memorising it separately doubles the material for no gain.
Myth 3: "A negative exponent can be left in the denominator." Truth: It can, but the minus sign must be tracked. Most lost points in this topic come from dropping it during the translation back.

Topic-Specific Study Method

Rewrite, Then Drop

You see	Rewrite	Derivative	Watch for
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$	slope infinite at $0$
$\frac{1}{\sqrt{x}}$	$x^{-1/2}$	$-\frac{1}{2}x^{-3/2}$	sign and domain
$\frac{5}{x^2}$	$5x^{-2}$	$-10x^{-3}$	coefficient survives
$x^2\sqrt{x}$	$x^{5/2}$	$\frac{5}{2}x^{3/2}$	add the exponents first
$\frac{x^3}{x}$	$x^2$	$2x$	simplify before differentiating

② Differentiate $\sqrt{x}$ three ways: by the rule, by the definition with a conjugate, and by reading the graph at $x = 4$ . All three should give $\frac{1}{4}$ .

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = 3\sqrt{x} - \frac{2}{x}$ .

Standard Execution:

Rewrite. $f(x) = 3x^{1/2} - 2x^{-1}$ .

Differentiate. $f'(x) = \frac{3}{2}x^{-1/2} + 2x^{-2}$ .

Translate back. $f'(x) = \frac{3}{2\sqrt{x}} + \frac{2}{x^2}$ .

Check. At $x = 1$ both forms give $3.5$ , and the original function is decreasing in its reciprocal term, which matches the positive second term.

Solution: Rewrite as $D(t) = 12(9-t)^{1/2}$ , so $D'(t) = 6(9-t)^{-1/2} \cdot (-1) = -\frac{6}{\sqrt{9-t}}$ . At $t = 5$ this gives $-3$ centimeters per minute. As $t \to 9^-$ the denominator tends to zero, so the rate becomes unbounded: the tank empties with a vertical tangent, which is the endpoint failure again.

Common Traps & Failure Signals:

The index as exponent (Signal: $\sqrt{x}$ differentiated to $\frac{1}{2}x^{1}$ or similar): the rewrite used the wrong power.
The lost sign (Signal: a positive answer where the function is decreasing): the negative exponent was translated back without its minus.
The unexamined endpoint (Signal: a finite slope reported at the start of a root): the domain was never checked.

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

Root Unrewritten: You never turned the root into a power.
Sign Lost $\rightarrow$ Carry the minus sign through the translation back.
Endpoint Ignored: You reported a finite slope at an input where the rewritten power has a negative exponent over a vanishing quantity, which is the formula's way of saying the tangent is vertical and no derivative exists there.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Fractional and negative exponents usually arrive already rewritten, and no released part from 2017 to 2026 scores the rewrite as an object of its own
10-Year Total Count: 1 of 9 Exams · 2 FRQ Parts; Both parts are 2019, and in both the rewritten power is the only route to the slope
FRQ Years Appeared: 2019 (Q4b, Q5a)

Practice Examples

MCQ 1

A Fractional Power

If $f(x) = x^{3/2}$ , what is $f'(x)$ ?

MCQ 2

A Reciprocal Root

If $g(x) = \frac{1}{\sqrt[3]{x}}$ , what is $g'(x)$ ?

MCQ 3

Why It Still Works

Why does the power rule apply to $x^{1/2}$ even though the exponent is not a whole number?

FRQ 1 • A Tank That Empties

Water drains from a tank so that the depth of water at time $t$ minutes is modeled by $D(t) = 12\sqrt{9 - t}$ centimeters for $0 \le t < 9$ .

(a) Write $D(t)$ as a power of $(9 - t)$ and find $D'(t)$ .

(b) Find the rate at which the depth is changing at $t = 5$ minutes. Include units.

(c) Explain why $D'(9)$ does not exist, using the form of $D'(t)$ rather than the graph.

(d) A second tank drains so that its depth is $E(t) = 12(9-t)^{1/3}$ . Compare $E'(5)$ with $D'(5)$ and explain which tank is draining faster at that instant.

Cheatsheet

Derivative of $x^n$

	Rewrite	Derivative
$x^n$	none	$nx^{n-1}$
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\sqrt[n]{x}$	$x^{1/n}$	$\frac{1}{n}x^{1/n - 1}$
$\frac{1}{x^n}$	$x^{-n}$	$-nx^{-n-1}$
$x^a x^b$	$x^{a+b}$	$(a+b)x^{a+b-1}$
$\frac{x^a}{x^b}$	$x^{a-b}$	$(a-b)x^{a-b-1}$

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Unit 2Derivative of

x^n

Concept 4.224Built for May 2027 AP Exam

Derivative of $x^n$

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A term with a radical, a reciprocal, or a power of a power, inside a rate, a tangent slope, or a second derivative.
What you’ll do: Convert to $x^n$ , differentiate with one step, then simplify back into the form the answer choices or the rubric expect.
Memory hook: A root is a fraction, a reciprocal is a negative. Costume first, rule second.
Use it on the exam: For MCQs: Options are frequently the same expression written two ways, one with a radical and one with a fractional exponent. Simplify back to the original costume before you compare. For FRQs: The rubric accepts a derivative in either form, but a later part usually substitutes a number. A negative exponent is easier to evaluate than a compound fraction, so convert toward whichever is easier to plug into.

Concept Review

Same Rule, Different Costumes

Three rows mapping a radical or reciprocal in its original form to a fractional or negative power and then to its derivative, showing that the power rule is unchanged once the rewrite is done

The Rewrites Worth Memorising

Written as	As a power	Derivative	Derivative, original form
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$	$\frac{1}{2\sqrt{x}}$
$\sqrt[3]{x}$	$x^{1/3}$	$\frac{1}{3}x^{-2/3}$	$\frac{1}{3\sqrt[3]{x^2}}$
$\frac{1}{x}$	$x^{-1}$	$-x^{-2}$	$-\frac{1}{x^2}$
$\frac{1}{x^4}$	$x^{-4}$	$-4x^{-5}$	$-\frac{4}{x^5}$
$x\sqrt{x}$	$x^{3/2}$	$\frac{3}{2}x^{1/2}$	$\frac{3\sqrt{x}}{2}$

Where the Rule Comes From

Expand $(x+h)^n$ with the binomial theorem and subtract $x^n$ :

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

Divide by $h$ and every term after the first still carries a factor of $h$ :

$\lim_{h \to 0}\frac{(x+h)^n - x^n}{h} = nx^{n-1} + 0 + \cdots = nx^{n-1}$

Where a Fractional Power Breaks

Common Misconception Check

Myth 1: "The power rule only applies to positive integer exponents." Truth: It applies to every constant exponent. The integer case is the easiest, not the only one.
Myth 2: "The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$ by a separate rule." Truth: It is the same rule after one rewrite. Memorising it separately doubles the material for no gain.
Myth 3: "A negative exponent can be left in the denominator." Truth: It can, but the minus sign must be tracked. Most lost points in this topic come from dropping it during the translation back.

Topic-Specific Study Method

Rewrite, Then Drop

You see	Rewrite	Derivative	Watch for
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$	slope infinite at $0$
$\frac{1}{\sqrt{x}}$	$x^{-1/2}$	$-\frac{1}{2}x^{-3/2}$	sign and domain
$\frac{5}{x^2}$	$5x^{-2}$	$-10x^{-3}$	coefficient survives
$x^2\sqrt{x}$	$x^{5/2}$	$\frac{5}{2}x^{3/2}$	add the exponents first
$\frac{x^3}{x}$	$x^2$	$2x$	simplify before differentiating

② Differentiate $\sqrt{x}$ three ways: by the rule, by the definition with a conjugate, and by reading the graph at $x = 4$ . All three should give $\frac{1}{4}$ .

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = 3\sqrt{x} - \frac{2}{x}$ .

Standard Execution:

Rewrite. $f(x) = 3x^{1/2} - 2x^{-1}$ .

Differentiate. $f'(x) = \frac{3}{2}x^{-1/2} + 2x^{-2}$ .

Translate back. $f'(x) = \frac{3}{2\sqrt{x}} + \frac{2}{x^2}$ .

Check. At $x = 1$ both forms give $3.5$ , and the original function is decreasing in its reciprocal term, which matches the positive second term.

Solution: Rewrite as $D(t) = 12(9-t)^{1/2}$ , so $D'(t) = 6(9-t)^{-1/2} \cdot (-1) = -\frac{6}{\sqrt{9-t}}$ . At $t = 5$ this gives $-3$ centimeters per minute. As $t \to 9^-$ the denominator tends to zero, so the rate becomes unbounded: the tank empties with a vertical tangent, which is the endpoint failure again.

Common Traps & Failure Signals:

The index as exponent (Signal: $\sqrt{x}$ differentiated to $\frac{1}{2}x^{1}$ or similar): the rewrite used the wrong power.
The lost sign (Signal: a positive answer where the function is decreasing): the negative exponent was translated back without its minus.
The unexamined endpoint (Signal: a finite slope reported at the start of a root): the domain was never checked.

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

Root Unrewritten: You never turned the root into a power.
Sign Lost $\rightarrow$ Carry the minus sign through the translation back.
Endpoint Ignored: You reported a finite slope at an input where the rewritten power has a negative exponent over a vanishing quantity, which is the formula's way of saying the tangent is vertical and no derivative exists there.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Fractional and negative exponents usually arrive already rewritten, and no released part from 2017 to 2026 scores the rewrite as an object of its own
10-Year Total Count: 1 of 9 Exams · 2 FRQ Parts; Both parts are 2019, and in both the rewritten power is the only route to the slope
FRQ Years Appeared: 2019 (Q4b, Q5a)

Practice Examples

MCQ 1

A Fractional Power

If $f(x) = x^{3/2}$ , what is $f'(x)$ ?

MCQ 2

A Reciprocal Root

If $g(x) = \frac{1}{\sqrt[3]{x}}$ , what is $g'(x)$ ?

MCQ 3

Why It Still Works

Why does the power rule apply to $x^{1/2}$ even though the exponent is not a whole number?

FRQ 1 • A Tank That Empties

Water drains from a tank so that the depth of water at time $t$ minutes is modeled by $D(t) = 12\sqrt{9 - t}$ centimeters for $0 \le t < 9$ .

(a) Write $D(t)$ as a power of $(9 - t)$ and find $D'(t)$ .

(b) Find the rate at which the depth is changing at $t = 5$ minutes. Include units.

(c) Explain why $D'(9)$ does not exist, using the form of $D'(t)$ rather than the graph.

(d) A second tank drains so that its depth is $E(t) = 12(9-t)^{1/3}$ . Compare $E'(5)$ with $D'(5)$ and explain which tank is draining faster at that instant.

Cheatsheet

Derivative of $x^n$

	Rewrite	Derivative
$x^n$	none	$nx^{n-1}$
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\sqrt[n]{x}$	$x^{1/n}$	$\frac{1}{n}x^{1/n - 1}$
$\frac{1}{x^n}$	$x^{-n}$	$-nx^{-n-1}$
$x^a x^b$	$x^{a+b}$	$(a+b)x^{a+b-1}$
$\frac{x^a}{x^b}$	$x^{a-b}$	$(a-b)x^{a-b-1}$

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On the AP Exam

Concept Review

Same Rule, Different Costumes

The Rewrites Worth Memorising

Where the Rule Comes From

Where a Fractional Power Breaks

Common Misconception Check

Topic-Specific Study Method

Rewrite, Then Drop

Training & Error Diagnosis

Past 10-Year AP Exam Appearance

Practice Examples

A Fractional Power

A Reciprocal Root

Why It Still Works

FRQ 1 • A Tank That Empties

Cheatsheet

Derivative of xnx^nxn

Ready to check your understanding?

On the AP Exam

Concept Review

Same Rule, Different Costumes

The Rewrites Worth Memorising

Where the Rule Comes From

Where a Fractional Power Breaks

Common Misconception Check

Topic-Specific Study Method

Rewrite, Then Drop

Training & Error Diagnosis

Past 10-Year AP Exam Appearance

Practice Examples

A Fractional Power

A Reciprocal Root

Why It Still Works

FRQ 1 • A Tank That Empties

Cheatsheet

Derivative of xnx^nxn

Ready to check your understanding?

On the AP Exam

Concept Review

Same Rule, Different Costumes

The Rewrites Worth Memorising

Where the Rule Comes From

Where a Fractional Power Breaks

Common Misconception Check

Topic-Specific Study Method

Rewrite, Then Drop

Training & Error Diagnosis

Past 10-Year AP Exam Appearance

Practice Examples

A Fractional Power

A Reciprocal Root

Why It Still Works

FRQ 1 • A Tank That Empties

Cheatsheet

Derivative of xnx^nxn

Ready to check your understanding?

Derivative of $x^n$

Derivative of $x^n$

Derivative of $x^n$