Constant Rule Explained — d/dx[c] = 0, Examples & Quizzes

Where it comes from

The graph of y = 7 is a perfectly flat horizontal line. Flat means zero rise over any run — zero slope everywhere. The derivative measures rate of change, and a constant never changes, so its rate of change must be 0. That is the whole intuition; the definition just confirms it.

The tempting wrong guess is that the derivative “keeps the number”: d/dx [5] = 5?? Kill it with the definition. For f(x) = 7:

Before reading on: for f(x) = 7, every difference quotient is (7 − 7)/h. What single number must the limit be — and what does that say about the graph’s slope?

[f(x+h) − f(x)] / h

=

(7 − 7) / h = 0

Every quotient is exactly 0, for every h ≠ 0. The limit of 0 is 0.

slope 5?

≠

flat line

A slope of 5 would mean the graph climbs 5 per unit of x — but the graph never moves at all. Dead on arrival.

This rule is the quiet workhorse of the chapter: it kills every stray number hanging off a polynomial, and it is the reason the derivative of x² + 9 is just 2x.

Derivation

Let f(x) = c, a constant. Apply the limit definition. This is the shortest proof in calculus:

f′(x)

=

lim_h→0 [f(x+h) − f(x)] / h

Step 1 — the definition. Nothing special yet.

=

lim_h→0 (c − c) / h

Step 2 — constants don’t move. f(x+h) = c and f(x) = c, because the function ignores its input entirely.

=

lim_h→0 0 = 0

Step 3 — done. 0/h = 0 for every h ≠ 0, and the limit of the constant 0 is 0. ∎

Bonus connection: the power rule with n = 0 gives d/dx [x⁰] = 0·x¹⁰¹ = 0 — and x⁰ = 1 (for x ≠ 0), so the constant rule is secretly the power rule in disguise.

How to use it

The procedure is one step: spot the constant, write 0. The skill is all in the spotting:

Ask: does it contain x? 7, π, e, √2, ln 5, π² — no x anywhere, so all constants. x², 5x, 2^x — contain x, so not constants.
Differentiate sums term by term (sum rule), and let the rule kill each constant term: d/dx [x² + 9] = 2x + 0 = 2x.
Don’t confuse it with the constant multiple rule. d/dx [5] = 0, but d/dx [5x] = 5 — the 5 multiplies x there, so it survives as a scaling factor.

Judgment calls

d/dx [ln 5] = 0. ln 5 ≈ 1.609 is just a number — the rule doesn’t care how fancy the number looks. d/dx [2^x] is NOT 0 — the x is in the exponent, so this needs the a^x rule (answer: 2^x ln 2).

Common mistake: writing d/dx [5x] = 0 (“the 5 is constant”). The 5 is constant, but it multiplies x — constant multiple rule, answer 5. Zero is only for lone constants.

Worked examples

Four problems, easiest first. The rule is one step — the lesson is in recognizing constants.

Example 1 — d/dx [π²]

Spot it: π² ≈ 9.87 is a number with no x. Constant.
Apply the rule: 0.

Common mistake: trying the power rule on the 2 (“bring down the 2”). The power rule differentiates xⁿ — powers of x. π² has no x, so there’s nothing to bring down.

Your turn: d/dx [e⁵]

Answer: 0

e⁵ ≈ 148.4 is a plain number with no x in it. Constant → 0. The e is decoration — the exponent 5 is fixed.

Example 2 — d/dx [x³ − 4]

Split the sum: d/dx [x³] − d/dx [4] (difference rule).
Power rule: d/dx [x³] = 3x².
Constant rule: d/dx [4] = 0.
Assemble: 3x². (Why does the −4 vanish? Shifting a graph down 4 doesn’t change any slope.)

Common mistake: answering 3x² − 4 — leaving the constant alive. Constants always die; only x-terms survive.

Your turn: d/dx [2x⁵ − 7]

Answer: 10x⁴

Split: d/dx [2x⁵] − d/dx [7] = 2·5x⁴ − 0 = 10x⁴. The −7 dies; the 2 multiplying x⁵ survives as a scale factor.

Example 3 — d/dx [e³ + √2]

Spot them: e³ ≈ 20.09 and √2 ≈ 1.414 — both plain numbers.
Both die: 0 + 0 = 0.

Common mistake: seeing e and reaching for d/dx [e^x] = e^x. That rule needs an x in the exponent — e³ is just a number.

Your turn: d/dx [ln 7 + π³]

Answer: 0

ln 7 ≈ 1.946 and π³ ≈ 31.0 are both plain numbers. 0 + 0 = 0 — fancy-looking constants are still constants.

Example 4 — the trap pair: d/dx [5] vs d/dx [5x]

d/dx [5]: lone constant → 0 (constant rule).
d/dx [5x]: 5 multiplies x → 5·d/dx [x] = 5 (constant multiple rule).

The lesson: ask whether the constant stands alone (dies) or multiplies x (survives as a scale factor). Five seconds of asking saves the mark.

Common mistake: answering 0 for d/dx [5x]. If that were right, the line y = 5x would be flat — but its slope is visibly 5.

Your turn: The trap pair: d/dx [−4] vs d/dx [−4x]

Answer: 0 and −4

d/dx [−4] = 0 (lone constant dies). d/dx [−4x] = −4·d/dx [x] = −4 (constant multiple rule — the −4 scales x, so it survives).

Memorization tips

Flat graph, flat slope: picture y = c as a tabletop. Zero tilt, zero derivative — the image is the rule.
The no-x test: cover the expression and ask “does x appear?” No x → 0. Works for π, e, √2, ln 5, even (2+3).
Constants die in sums: d/dx [x² + 9] = 2x. Read polynomials as “x-stuff survives, lonely numbers die.”
Alone vs. multiplying: d/dx [5] = 0 but d/dx [5x] = 5. One word — alone — decides which rule fires.
Fancy numbers are still numbers: d/dx [π²] = 0, d/dx [e³] = 0, d/dx [ln 5] = 0. Don’t let notation scare you into other rules.
Collapse check: any rule applied to a constant must give 0. If your product-rule expansion of d/dx [3x] doesn’t simplify to 3, something’s off.

Final challenge

Five mixed questions — spotting constants, dodging the 5x trap. Score 5/5 and the rule is yours.

How to learn a formula here

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.
Work the examples with the answers covered, then uncover one step at a time and compare.
Finish with the final challenge — five mixed questions including the classic traps.
Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

What is the constant rule?

The constant rule says d/dx[c] = 0: the derivative of any constant — any number or expression with no x in it — is zero, because a constant never changes.

Why isn’t the derivative of a constant the constant itself?

A slope of c would mean the graph climbs c units per unit of x. But the graph of y = c is perfectly flat — it never climbs at all — so its slope is 0. The limit definition confirms it: (c−c)/h = 0 for every h.

Is d/dx[π] really 0?

Yes. π ≈ 3.14159 is just a number with no x in it, so it’s a constant and its derivative is 0 — same for e, √2, π², and ln 5.

What’s the difference between the constant rule and the constant multiple rule?

The constant rule kills lone constants: d/dx[5] = 0. The constant multiple rule scales: d/dx[5x] = 5·d/dx[x] = 5 — the constant survives as a multiplier because x is still changing underneath it.

Does d/dx[2ˣ] = 0?

No — that’s the trap. 2ˣ contains x (in the exponent), so it isn’t constant; it needs the aˣ rule: d/dx[2ˣ] = 2ˣ·ln 2. The constant rule only fires when x appears nowhere.

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