Unit 2Power Rule Formula
Concept 4.225Built for May 2027 AP Exam

Power Rule Formula

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The power rule is one line of algebra that replaces every limit you would otherwise compute: ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1} for any constant nn. The exponent comes down as a multiplier and then decreases by one. Constants vanish, a lone xx becomes 11, and a coefficient rides along untouched. Applied term by term, the rule differentiates every polynomial instantly. The mistakes are equally mechanical: keeping the old exponent, forgetting the coefficient multiplier, or differentiating a constant into itself. On the BC papers the rule is rarely named, because released free-response parts usually hand you f′f' or a table instead of a formula. It still sits underneath every tangent slope, rate, and critical point you will compute from a given expression.

On the AP Exam

What you’ll see
A polynomial or a single power given as a formula, followed by a request for a tangent slope, a critical point, a rate of change, or a second derivative.
What you’ll do
Rewrite anything hidden (radicals, fractions, negative powers), then apply nxn−1nx^{n-1} term by term and simplify before substituting numbers.
Memory hook
Down, then one less. The exponent falls into the coefficient seat, and the seat gets one smaller.
Use it on the exam
For MCQs: The distractor set is built from the three mechanical slips: the old exponent kept, the decrement skipped, and the coefficient not multiplied. Differentiate twice on scrap paper when an option looks familiar. For FRQs: The derivative is usually one step inside a larger part, so it earns no point of its own. That makes speed matter more than presentation: write p′(h)p'(h) in one line, then move to the substitution the part is actually paying for.

Concept Review

Two Moves, No Limit

Figure 1 — The exponent performs two jobs in one step: it becomes the coefficient, and it is replaced by one less. The three white chips apply that to a coefficient, a bare xx, and a constant, and the amber chip is the error that keeps the original exponent.

The power rule shown as a before and after on one term with the exponent becoming a factor and dropping by one, followed by three worked terms and one crossed-out error where the exponent never dropped

The Rule and Its Edges

ddxxn=nxn−1for every constant n\frac{d}{dx}x^n = nx^{n-1} \qquad \text{for every constant } n

ExpressionRewrite firstDerivative
x7x^7none7x67x^6
xxx1x^111
555x05x^000
x\sqrt{x}x1/2x^{1/2}12x−1/2\frac{1}{2}x^{-1/2}
1x3\frac{1}{x^3}x−3x^{-3}−3x−4-3x^{-4}

The last two rows are the ones that cost points, and not because the rule changes: it does not. What changes is that the expression has to be rewritten into the form xnx^n before the rule can see it.

Term by Term on a Polynomial

A polynomial is a sum, and differentiation passes through sums and constant multiples without alteration. That is why one rule differentiates

f(x)=4x5−3x2+7x−9⟹f′(x)=20x4−6x+7f(x) = 4x^5 - 3x^2 + 7x - 9 \quad\Longrightarrow\quad f'(x) = 20x^4 - 6x + 7

in a single line. Each term is handled alone: 4x54x^5 gives 20x420x^4, −3x2-3x^2 gives −6x-6x, 7x7x gives 77, and −9-9 gives 00. The linear term is the one students drop by accident, because "7x7x becomes 77" does not look like the rule they memorised.

A Check That Catches Most Slips

Every differentiated term can be audited with two questions. Is the new exponent exactly one less than the old one? Is the coefficient exactly the old coefficient times the old exponent? Both answers must be yes, and neither check needs any algebra. On 4x54x^5 the audit reads: the coefficient should be 4⋅5=204 \cdot 5 = 20 and the exponent should be 5−1=45 - 1 = 4, which matches 20x420x^4. Applied to a whole polynomial in ten seconds, this catches the kept exponent, the missing multiplier, and the surviving constant.

Common Misconception Check

  • Myth 1: "The exponent stays the same and only the coefficient appears." Truth: Both moves happen. The coefficient is nn and the new exponent is n−1n-1.
  • Myth 2: "Constants differentiate to themselves." Truth: A constant has slope zero. Only the coefficient of a variable term survives.
  • Myth 3: "The rule works only for positive whole numbers." Truth: It holds for every constant nn, which is what makes radicals and reciprocals cheap to differentiate once they are rewritten.

Topic-Specific Study Method

Rewrite, Then Drop

Borrowed idea: Where does the exponent go? It becomes the coefficient, and the exponent itself drops by one. Keep those two moves straight and the rule never misleads you. Constant terms disappear, because a flat line has no slope. A lone xx becomes 11, because the exponent was there all along. A term like 3x43x^4 keeps its coefficient and multiplies it by 4 on the way down. Do you need to rewrite anything first? Only when the power is hidden inside a radical or a fraction, and then the rewrite belongs before the differentiation. Does the rule care about the coefficient? No, it passes through untouched. Work term by term, and check each exponent by asking what one less than it is.

You seeRewrite asThen differentiate
x\sqrt{x}x1/2x^{1/2}12x−1/2\frac{1}{2}x^{-1/2}
3x2\frac{3}{x^2}3x−23x^{-2}−6x−3-6x^{-3}
xxx\sqrt{x}x3/2x^{3/2}32x1/2\frac{3}{2}x^{1/2}
x2+xx\frac{x^2 + x}{x}x+1x + 111
777x07x^000

① Differentiate f(x)=x6−4x3+2x−11f(x) = x^6 - 4x^3 + 2x - 11 twice, then three times. Watch the polynomial shorten and then vanish, and notice that each pass applies the same two moves.

② Rewrite five radicals and five reciprocals from your practice set as powers before differentiating any of them. The rewrite step is where the marks are lost, not the differentiation.

③ Check every answer by asking what one less than the new exponent is. If the new exponent plus one does not equal the exponent you started with, the term was misread.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the slope of the line tangent to f(x)=2x4−5x2+3f(x) = 2x^4 - 5x^2 + 3 at x=1x = 1.

Standard Execution:

  1. Differentiate. f′(x)=8x3−10xf'(x) = 8x^3 - 10x.
  2. Substitute. f′(1)=8−10=−2f'(1) = 8 - 10 = -2.
  3. Check the point. f(1)=2−5+3=0f(1) = 2 - 5 + 3 = 0.
  4. Report. The tangent line is y−0=−2(x−1)y - 0 = -2(x - 1), so the slope is −2-2.

Exam Variation (Context / Parameter Shift): A particle moves along a line with position s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9t for t≥0t \ge 0, in meters and seconds. Find every time at which the particle is at rest, and determine when it is moving in the negative direction.

Solution: v(t)=s′(t)=3t2−12t+9=3(t−1)(t−3)v(t) = s'(t) = 3t^2 - 12t + 9 = 3(t-1)(t-3), so the particle is at rest at t=1t = 1 and t=3t = 3. Testing the intervals gives v>0v > 0 on [0,1)[0,1), v<0v < 0 on (1,3)(1,3), and v>0v > 0 for t>3t > 3, so the particle moves in the negative direction only on 1<t<31 < t < 3. The power rule produced the velocity; the sign chart did the rest.

Common Traps & Failure Signals:

  1. The exponent kept (Signal: x5x^5 differentiated to 5x55x^5): the decrement was skipped.
  2. The constant survived (Signal: a +7+7 still present in the derivative): a constant was treated as a term of degree zero with slope one.
  3. The hidden power (Signal: a radical differentiated without a rewrite): the expression was never put into the form xnx^n.

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

  • Drop Missed →\rightarrow Say the new exponent out loud first.
  • Constant Kept: You differentiated a constant into itself, which happens when the rule is recalled as a pattern of symbols rather than as a statement about slopes.
  • Rewrite Skipped: You applied the rule to a radical without converting it to a fractional power, so the exponent you dropped was the index of the root rather than the exponent of the power.

Past 10-Year AP Exam Appearance

Testing Frequency
Rare in FRQs
The rule is assumed rather than examined: where a released part needs an integer power differentiated, the power sits inside a product or a power of a trig function, so the rule is one step rather than the object of the part
10-Year Total Count
3 of 9 Exams · 3 FRQ Parts
Counted only where the scored work cannot be finished without differentiating an integer power
FRQ Years Appeared
2018 (Q2a), 2021 (Q2a), 2023 (Q1d)

Practice Examples

MCQ 1

One Term

If f(x)=6x5f(x) = 6x^5, what is f′(x)f'(x)?

MCQ 2

A Polynomial

If g(x)=3x4−2x2+9g(x) = 3x^4 - 2x^2 + 9, what is g′(x)g'(x)?

MCQ 3

Before Differentiating

If h(x)=4x2h(x) = \frac{4}{x^2}, what is h′(x)h'(x)?

FRQ 1 • Position, Velocity, and the Power Rule

A particle moves along the xx-axis with position s(t)=t4−8t2+16s(t) = t^4 - 8t^2 + 16 for t≥0t \ge 0, where ss is measured in meters and tt in seconds.

(a) Find v(t)v(t), the velocity of the particle.

(b) Find every time t>0t > 0 at which the particle is at rest. Show the work that leads to your answer.

(c) Find the acceleration a(t)a(t), then determine the acceleration at t=2t = 2 seconds and state its units.

(d) On which time intervals is the particle moving to the left? Justify your answer with a sign analysis of vv.

Cheatsheet

Power Rule

RewriteDerivative
xnx^nnonenxn−1nx^{n-1}
cxncx^nnonecnxn−1cnx^{n-1}
cccx0cx^000
xxx1x^111
x\sqrt{x}x1/2x^{1/2}12x−1/2\frac{1}{2}x^{-1/2}
1xn\frac{1}{x^n}x−nx^{-n}−nx−n−1-nx^{-n-1}
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