Calculus I › Differentiation rules › full formula sheet

d/dx [xn] = n·xn−1Say it: the derivative of x to the n is n times x to the n minus one.

The power rule

The single most-used rule in calculus: bring the exponent down, drop it by one — and here is why that actually works.

Notation: n is any fixed real number (2, −3, 1/2, π — anything). The rule needs a variable base x with a constant exponent.

Before this lesson: Definition of the derivative · Constant rule

Where it comes from

Before the power rule, every xn needed the full limit definition — minutes of algebra per derivative. The pattern was begging to be bottled: x² → 2x, x³ → 3x². The guess writes itself: bring the exponent down front, then drop it by one.

But the lazy guess is “bring it down and keep it”: d/dx [xn] = n·xn?? Kill it with n = 2, x = 3:

Before reading on: at n = 2, x = 3 the definition gives slope 6. Which candidate fits — n·xn or n·xn−1 — and what does the loser predict?

true (definition)
=
2 · 3 = 6
We computed ((x+h)²−x²)/h → 2x on the definition page; at x = 3 that’s 6.
lazy guess nxn
=
2 · 3² = 18
Off by a factor of 3. The exponent must drop — 18 ≠ 6, dead on arrival.

Why does it drop by exactly one? Expand (x+h)n: you get xn, plus n·xn−1·h, plus terms with h², h³, …. Subtract xn, divide by h, and only the one-h term survives the limit — every higher term still carries an h and dies. Dividing by h once eats exactly one power: that’s the −1.

Derivation

For a positive integer n, the binomial theorem expands (x+h)n. Watch which terms survive:

Before reading on: expand (x+h)n. After subtracting xn and dividing by h, exactly one term has no h left. Which is it — and why must every other term die in the limit?

d/dx [xn]
=
limh→0 [(x+h)n − xn] / h
Step 1 — the definition.
=
limh→0 [xn + n·xn−1h + (…h² terms…) − xn] / h
Step 2 — binomial expansion. (x+h)n = xn + n·xn−1·h + terms each containing h² or higher. The xn cancels.
=
limh→0 [n·xn−1 + h·(stuff)]
Step 3 — divide by h. The one-h term becomes n·xn−1; every remaining term still has a factor of h.
=
n·xn−1
Step 4 — the limit. h·(stuff) → 0. Only n·xn−1 stands. ∎

Beyond integers: this proof covers n = 1, 2, 3, …. The rule also holds for every real n — negative, fractional, even π — proved later using logarithms (take ln of both sides). You may use it freely for all real n.

How to use it

The procedure, every time:

  1. Rewrite first. Roots and fractions must become powers: √x = x1/2, ∛x = x1/3, 1/x³ = x−3. The rule only sees xn form.
  2. Bring down: multiply by n. Drop: new exponent n−1.
  3. Clean up: negative exponents become fractions (x−4 = 1/x⁴); fractional exponents can go back to roots.
  4. Check the edges: n = 1 gives 1·x⁰ = 1 (the line y = x has slope 1 ✓); n = 0 gives 0 (the constant rule ✓).

When NOT to use it

The power rule needs a variable base and constant exponent. d/dx [2x] has a constant base — that’s the ax rule (2x ln 2). d/dx [xx] has neither fixed — that needs logarithmic differentiation, a later topic.

Common mistake: differentiating 1/x² as if the exponent were +2, giving 2/x or 2x. Rewrite FIRST: x−2 → −2x−3 = −2/x³. The minus sign on the exponent is load-bearing.

Worked examples

Four problems, easiest first. In each one, the rewrite step is half the battle.

Example 1 — d/dx [x⁷]

  1. Bring down: 7. Drop: 7−1 = 6.
  2. Answer: 7x⁶. (Why believe it? At x = 1 the slope is 7 — matches the definition computation.)
Common mistake: writing 7x⁷ (forgetting to drop). The n = 2, x = 3 test kills it: 2·3² = 18 ≠ 6.
Your turn: d/dx [x⁷]

Answer: 9x⁸

Bring down 9, drop to 8: 9x⁸. Check at x = 1: slope 9 — matches (x+h)⁷ expanded near 1.

Example 2 — d/dx [√x] (rewrite first!)

  1. Rewrite: √x = x1/2.
  2. Bring down 1/2, drop to −1/2: (1/2)·x−1/2.
  3. Clean up: = 1/(2√x).
  4. Sanity check: at x = 4 the slope is 1/4 — small and positive, matching the flattening root curve. ✓
Common mistake: answering (1/2)√x or (1/2)x1/2 — forgetting the drop turns the exponent negative. Fractional exponents still drop by one: 1/2 − 1 = −1/2.
Your turn: d/dx [1/√x] (rewrite first!)

Answer: −1/(2x3/2)

Rewrite: 1/√x = x−1/2. Bring down −1/2, drop to −3/2: −(1/2)x−3/2 = −1/(2x3/2).

Example 3 — d/dx [1/x²] (negative exponents)

  1. Rewrite: 1/x² = x−2.
  2. Bring down −2, drop to −3: −2·x−3.
  3. Clean up: = −2/x³.
  4. Check via quotient rule: [(0)(x²) − 1·(2x)]/x⁴ = −2x/x⁴ = −2/x³. Matches ✓
Common mistake: losing the minus: writing 2/x³. The exponent was negative, so the multiplier is negative — track signs through the rewrite.
Your turn: d/dx [2/x³] (negative exponents)

Answer: −6/x⁴

Rewrite: 2/x³ = 2x−3. Bring down −3, drop to −4: 2·(−3)x−4 = −6x−4 = −6/x⁴.

Example 4 — d/dx [x⁵ − 4x−1 + ∛x] (mixed)

  1. Rewrite the root: ∛x = x1/3. Term by term (sum rule):
  2. x⁵ → 5x⁴.
  3. −4x−1 → −4·(−1)·x−2 = 4/x². (Why positive? Minus times minus.)
  4. x1/3 → (1/3)x−2/3 = 1/(3∛(x²)).
  5. Assemble: 5x⁴ + 4/x² + 1/(3∛(x²)).
Common mistake: differentiating −4x−1 as −4x−2 (dropping the exponent without bringing it down). Both moves — bring down and drop — happen every time.
Your turn: d/dx [3x⁴ + 2/x − √x] (mixed)

Answer: 12x³ − 2/x² − 1/(2√x)

Rewrite: 3x⁴ + 2x−1 − x1/2. Term by term: 12x³ − 2x−2 − (1/2)x−1/2 = 12x³ − 2/x² − 1/(2√x).

Memorization tips

  • Chant it: “down in front, one off the top.” Two beats, two moves — say it while you write.
  • Rewrite before you rule: √x, 1/x³, ∛(x²) — convert everything to xn form first. Ninety percent of power-rule errors are skipped rewrites.
  • The n = 2, x = 3 test: 2·3 = 6. Run any doubtful formula through it — the lazy nxn guess gives 18 and dies instantly.
  • Fractional n still drops by one: 1/2 − 1 = −1/2, 2/3 − 1 = −1/3. Do the subtraction carefully; it’s the #1 arithmetic slip.
  • Negative n gives negative multipliers: d/dx [x−3] = −3x−4. If your answer lost a minus, the rewrite step is where it went.
  • Edge cases are free checks: n = 1 gives 1, n = 0 gives 0. If your “general” method fails these, the method is wrong, not the edges.

Final challenge

Five mixed questions — fractional exponents, the xˣ trap, and speed. Score 5/5 and the rule is yours.

← Back to the Calculus I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

What is the power rule?

The power rule says d/dx[xⁿ] = n·xⁿ⁻¹: multiply by the exponent, then subtract one from the exponent. It works for any fixed real exponent n — positive, negative, or fractional.

Why does the exponent drop by one?

Expanding (x+h)ⁿ gives a term n·xⁿ⁻¹·h plus higher-h terms. Dividing the difference quotient by h eats one factor of h, and the limit kills the rest — so exactly one power is lost.

Do I need to rewrite square roots before using the power rule?

Yes — the rule only recognizes xⁿ form. Write √x = x^(1/2), ∛x = x^(1/3), and 1/x³ = x^−3 first, then bring down and drop.

Can I use the power rule on 2ˣ?

No. The power rule needs a variable base with a constant exponent. 2ˣ has a constant base and variable exponent — use the aˣ rule: d/dx[2ˣ] = 2ˣ·ln 2.

Does the power rule work for fractional and negative exponents?

Yes — for every real n. Examples: d/dx[x^(1/2)] = (1/2)x^(−1/2) and d/dx[x^−2] = −2x^−3. The integer proof extends via logarithms.

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