Constant Multiple Rule Explained — d/dx[c·f] = c·f′ & Quizzes

Where it comes from

Compare f(x) = x² with g(x) = 3x². The second graph is the first stretched vertically by 3 — every y-value tripled. A vertical stretch by 3 triples every rise, and slope is rise over run, so every slope triples too. At x = 1: f has slope 2, g has slope 6 = 3·2.

The wrong instinct is to “differentiate the constant too” or to drop it. Test the drop: if d/dx [3x²] were just 2x, then at x = 1 the slope would be 2 — but the stretched graph is visibly 3× steeper than x². Dropping the 3 contradicts the picture:

Before reading on: g(x) = 3x² is f(x) = x² stretched vertically by 3. At x = 1, f has slope 2 — what slope would you predict for g, and what does that say about the 3?

slope of x² at x=1

=

2

From the power rule: 2x at x = 1.

slope of 3x² at x=1

=

6 = 3 · 2

Stretch by 3 → slopes ×3. The constant rides along untouched.

Think of c as a fixed zoom factor: it scales the output, so it scales every rate of change by the same factor. The rule just says: pull the zoom out front, differentiate the shape.

Derivation

Let F(x) = c·f(x) with f differentiable at x. Apply the definition — the constant factors straight out of the limit:

F′(x)

=

lim_h→0 [c·f(x+h) − c·f(x)] / h

Step 1 — the definition applied to F = c·f.

=

lim_h→0 c · [f(x+h) − f(x)] / h

Step 2 — factor c out. c doesn’t depend on h, so it pulls out of the numerator (and out of the limit next).

=

c · lim_h→0 [f(x+h) − f(x)] / h

Step 3 — the constant multiple law for limits. A constant factor slides out of any limit.

=

c · f′(x)

Step 4 — recognize the definition. The remaining limit is f′(x). ∎

Collapse check: if c = 0, the rule gives 0·f′(x) = 0 — exactly the constant rule for the function F(x) = 0. The rules agree where they overlap.

How to use it

The procedure: pull the constant out front, differentiate the rest.

Identify the constant factor — the number multiplying the whole x-expression: 4 in 4x³, −2 in −2 sin x, 1/3 in (1/3)x².
Write it outside the derivative: d/dx [4x³] = 4·d/dx [x³].
Differentiate what remains with its own rule: 4·3x² = 12x².
Works for every constant — negative (flip the sign), fractional (keep the fraction), π (just rides along).

Don’t reach for the product rule

d/dx [3x²] could be done as a product: (3)′·x² + 3·(x²)′ = 0 + 6x = 6x. Same answer, triple the writing — and every extra line is a chance to slip. The constant multiple rule always beats the product rule on c·f.

Common mistake: “differentiating the constant” — writing d/dx [4x²] = 4·2x + something, or turning the 4 into 0 and killing the term. The constant is a passenger: it neither differentiates nor disappears, it just rides along.

Worked examples

Four problems, easiest first. In each one, the first move is pulling the constant out.

Example 1 — d/dx [4x³]

Pull out the 4: = 4·d/dx [x³].
Power rule: d/dx [x³] = 3x².
Multiply: = 12x².

Common mistake: writing 4·3x² as 12x³ (forgetting the power rule’s drop). The constant rides along; the exponent still drops.

Your turn: d/dx [6x⁵]

Answer: 30x⁴

Pull out the 6: 6·d/dx [x⁵] = 6·5x⁴ = 30x⁴. The constant rides along; the exponent still drops.

Example 2 — d/dx [−2 sin x] (negative constant)

Pull out the −2: = −2·d/dx [sin x].
Sine rule: d/dx [sin x] = cos x.
Multiply: = −2 cos x. (Why negative? The −2 flips every slope.)

Common mistake: dropping the minus and answering 2 cos x. Carry the sign with the constant from line one.

Your turn: d/dx [3 cos x]

Answer: −3 sin x

Pull out the 3: 3·d/dx [cos x] = 3·(−sin x) = −3 sin x. Cosine’s minus survives the ride.

Example 3 — d/dx [5/x] (rewrite first)

Rewrite: 5/x = 5x⁻¹ — now the 5 is visibly a constant factor.
Pull out the 5: = 5·d/dx [x⁻¹].
Power rule: d/dx [x⁻¹] = −1·x⁻².
Multiply and tidy: = −5/x².

Common mistake: reaching for the quotient rule on 5/x. It works — [(0)(x) − 5(1)]/x² — but the rewrite + constant multiple is two lines instead of five.

Your turn: d/dx [7/x²] (rewrite first)

Answer: −14/x³

Rewrite: 7/x² = 7x⁻². Pull out the 7: 7·d/dx [x⁻²] = 7·(−2x⁻³) = −14x⁻³ = −14/x³.

Example 4 — d/dx [(1/3)x² + 7x] (fraction + sum)

Split the sum: = d/dx [(1/3)x²] + d/dx [7x].
First term: (1/3)·d/dx [x²] = (1/3)·2x = (2/3)x. (Why keep the fraction? (1/3)·2 = 2/3 — no decimal needed.)
Second term: 7·d/dx [x] = 7·1 = 7.
Assemble: (2/3)x + 7.

Common mistake: writing the first term as (2/3)x² (bringing down without dropping) or 2x (losing the 1/3). Fractions ride along exactly like integers.

Your turn: d/dx [(2/5)x³ − 4x] (fraction + sum)

Answer: (6/5)x² − 4

Split the sum. First term: (2/5)·3x² = (6/5)x². Second term: 4·1 = 4. Assemble: (6/5)x² − 4.

Memorization tips

Passenger, not driver: the constant never differentiates and never disappears — it rides along outside the derivative. Say “pull it out front” as you write.
Stretch picture: c = 3 triples every slope, c = −1 flips them, c = 1/2 halves them. The sign and size of c tell you what happened to the graph.
Always beats the product rule: d/dx [c·f] via product rule gives 0·f + c·f′ — the same answer with extra steps. Skip the theater.
Rewrite to reveal: 5/x hides its constant factor; 5x⁻¹ shows it. Rewrite first, then pull out.
Chain it with sums: d/dx [3x² + 2x] = 3·2x + 2·1. Pull each constant out of its own term.
Collapse check: c = 0 gives 0 — matching the constant rule. c = 1 gives back f′ untouched. If your use fails these, recheck.

Final challenge

Five mixed questions — negatives, fractions, and the product-rule temptation. 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 multiple rule?

The constant multiple rule says d/dx[c·f(x)] = c·f′(x): a constant factor pulls out in front of the derivative untouched. Stretching a graph vertically by c stretches every slope by c.

Why does the constant survive instead of differentiating to 0?

Because it multiplies something that changes. In d/dx[3x²], the 3 scales x²’s growth — it’s not sitting alone. Only lone constants (d/dx[3] = 0) differentiate to zero.

Can the constant be negative or a fraction?

Yes — any fixed number. d/dx[−2·sin x] = −2·cos x and d/dx[(1/3)x²] = (2/3)x. Negatives flip the slopes’ signs; fractions shrink them.

Should I use the product rule on 4x³ instead?

You could, but don’t: (4)′·x³ + 4·(x³)′ = 0 + 12x² is the same answer with wasted steps. The constant multiple rule is the shortcut — that’s why it exists.

How is this different from the constant rule?

The constant rule kills lone constants: d/dx[5] = 0. The constant multiple rule preserves multipliers: d/dx[5x] = 5. Ask: does the constant stand alone (dies) or multiply x (rides along)?

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