Calculus I › Differentiation rules › full formula sheet
The sum / difference rule
Rates add: differentiate each piece with its own rule and add the results — the workhorse behind every polynomial derivative.
Notation: f and g are differentiable functions of x. ± means the sign carries through: plus stays plus, minus stays minus.
Before this lesson: Definition of the derivative · Constant multiple rule
Where it comes from
If f grows at 2 units per second and g grows at 3 units per second, then f+g grows at 5 units per second — rates add. The derivative of a sum is the sum of the derivatives because change doesn’t care how you grouped the terms.
The wrong guess to kill: (f+g)′ = f′·g′?? Take f(x) = g(x) = x:
Before reading on: if f grows at 2 units per second and g at 3 units per second, how fast does f+g grow — and what does that make (f+g)′ in terms of f′ and g′?
The intuition is even simpler than the product rule’s rectangle: a sum’s change is just each part’s change, added. No cross terms, no tricks — which is exactly why polynomials differentiate term by term.
Derivation
Let S(x) = f(x) + g(x) with f, g differentiable at x. The limit definition splits cleanly:
For differences, the same steps carry the minus through:
And it extends to any number of terms: (f+g−k)′ = f′ + g′ − k′.
How to use it
The procedure: split the sum, differentiate each piece with its own rule, recombine.
- Split into terms: d/dx [x³ + 5x − 7] becomes three separate derivatives.
- Differentiate each term with whatever rule it needs: power rule for x³, constant multiple for 5x, constant rule for 7.
- Recombine with the original signs: 3x² + 5 − 0 = 3x² + 5.
- Watch subtraction signs. d/dx [x² − 3x] = 2x − 3 — the minus survives. This is the #1 sign error in the chapter.
Judgment calls
Simplify first when it’s free: d/dx [(x²+1) − (x²−1)] is 2x − 2x = 0 by the rule — but simplifying the inside to 2 first gives d/dx [2] = 0 in one step. Don’t split products or compositions: x² sin x needs the product rule, sin(x²) the chain rule — the sum rule only splits + and −.
Worked examples
Four problems, easiest first. Each term gets its own rule — that’s the whole game.
Example 1 — d/dx [x³ + 5x]
- Split: d/dx [x³] + d/dx [5x].
- Power rule: 3x². Constant multiple: 5·1 = 5.
- Assemble: 3x² + 5.
Your turn: d/dx [x⁴ − 3x²]
Answer: 4x³ − 6x
Split: d/dx [x⁴] − d/dx [3x²] = 4x³ − 3·2x = 4x³ − 6x. Keep the minus on the second term.
Example 2 — d/dx [x² − √x] (sign + rewrite)
- Split (keep the minus): d/dx [x²] − d/dx [√x].
- Rewrite the root: √x = x1/2 → (1/2)x−1/2 = 1/(2√x).
- Assemble: 2x − 1/(2√x).
Your turn: d/dx [√x + 1/x] (rewrite)
Answer: 1/(2√x) − 1/x²
Rewrite: x1/2 + x−1. Differentiate term by term: (1/2)x−1/2 − x−2 = 1/(2√x) − 1/x².
Example 3 — d/dx [4x⁵ − 3x² + 7x − 1] (full polynomial)
- Term by term: 4·5x⁴ − 3·2x + 7·1 − 0.
- Multiply out: 20x⁴ − 6x + 7 − 0.
- Assemble: 20x⁴ − 6x + 7. (Why did −1 die? Constant rule: lone numbers differentiate to 0.)
Your turn: d/dx [2x⁶ + 5x³ − 8x + 3] (full polynomial)
Answer: 12x⁵ + 15x² − 8
Term by term: 2·6x⁵ + 5·3x² − 8·1 + 0 = 12x⁵ + 15x² − 8. The lone 3 dies by the constant rule.
Example 4 — d/dx [sin x + cos x] (transcendental)
- Split: d/dx [sin x] + d/dx [cos x].
- Trig rules: cos x + (−sin x). (Why the minus? The derivative of cosine is negative sine.)
- Assemble: cos x − sin x.
Your turn: d/dx [sin x − cos x] (transcendental)
Answer: cos x + sin x
Split: cos x − (−sin x) = cos x + sin x. The difference rule’s minus meets cosine’s minus — minus times minus.
Memorization tips
- Term-by-term is the whole rule: see a polynomial, split it mentally into columns and differentiate each. The rule turns one scary problem into easy ones.
- Copy the signs: plus stays plus, minus stays minus. Write the sign before the derivative so subtraction can’t sneak away.
- Each term brings its own rule: x³ wants power, 5x wants constant multiple, 7 wants constant. The sum rule is the dispatcher, not the worker.
- Constants die, multipliers survive: in x² + 9 the 9 dies; in 9x the 9 survives as 9. Alone vs. multiplying — same test as the constant pages.
- Simplify-then-split: (x²+1)−(x²−1) = 2 collapses before you differentiate. Look for free simplifications first.
- Only splits + and −: products need the product rule, compositions the chain rule. If there’s no + or − on top, this rule isn’t the one.
Final challenge
Five mixed questions — sign traps, multi-term polynomials, and the product lookalike. Score 5/5 and the rule is yours.
← Back to the Calculus I formula sheet
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 sum/difference rule?
The sum/difference rule says d/dx[f ± g] = f′ ± g′: differentiate each piece with its own rule and add (or subtract) the results. Rates add — that’s all it is.
Why does the minus sign survive in (f−g)′ = f′−g′?
Because the proof regroups the numerator as f’s quotient minus g’s quotient — the subtraction is structural, not cosmetic. Dropping it claims g’s change adds to f’s, which is false.
Can I split more than two terms?
Yes — apply the rule repeatedly: (f+g−k)′ = f′+g′−k′. That’s how every polynomial differentiates term by term.
Does the sum rule work for products?
No — (fg)′ = f′g + fg′, not f′+g′. Counterexample: f(x) = x², g(x) = x. The product is x³, so (fg)′ = 3x²; but f′+g′ = 2x+1. Since 3x² ≠ 2x+1, products need the product rule.
What’s the most common sum-rule mistake?
Sign errors in subtraction: writing d/dx[x²−x] = 2x+1 instead of 2x−1. Copy each term’s sign onto its derivative before computing anything.
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