Calculus I › Differentiation rules › full formula sheet

d/dx [cos x] = −sin xSay it: the derivative of cosine of x is negative sine of x.

The derivative of cos x

Cosine’s derivative is negative sine — the graph demands the minus, and the limit proof delivers it.

Notation: x is in radians. d/dx[cos(u)] = −sin(u)·u′ when there’s something inside. Chant: “sine to cosine, cosine to negative sine.”

Before this lesson: Derivative of sin x · Definition of the derivative

Where it comes from

Picture the cosine wave. It starts at its peak (1, 0) and immediately heads down — negative slope from the start. At x = π/2 it plunges through zero at 45°: slope −1. Which candidate fits? −sin(π/2) = −1, while +sin(π/2) = +1. The graph demands the minus — the no-minus guess dies at π/2.

The shift picture works here too: cosine’s slope pattern (0 at 0, −1 at π/2, 0 at π, +1 at 3π/2) traces a wave that is sine flipped upside down — negative sine:

Before reading on: cosine plunges through x = π/2 heading down. What slope does the graph demand there — and which of sin x / −sin x delivers it?

slope of cos at π/2
=
−1 = −sin(π/2)
Plunging through zero — the minus is mandatory.
slope of cos at 0
=
0 = −sin(0)
At the peak the tangent is flat — zero slope.

Memory anchor: sine is the plus one, cosine is the minus one. The asymmetry comes from where each wave starts — sine starts climbing (positive slope), cosine starts falling (negative slope).

Derivation

Apply the definition to f(x) = cos x, using cos(x+h) = cos x cos h − sin x sin h — note the minus in the formula, the seed of our answer’s minus:

d/dx [cos x]
=
limh→0 [cos(x+h) − cos x] / h
Step 1 — the definition.
=
limh→0 [cos x cos h − sin x sin h − cos x] / h
Step 2 — angle-sum formula. Expand cos(x+h).
=
limh→0 [ cos x · (cos h−1)/h  −  sin x · sin h/h ]
Step 3 — regroup. The angle-sum’s minus lands in front of sin x’s group — this is where the answer’s minus is born.
=
cos x · 0  −  sin x · 1 = −sin x
Step 4 — the two key limits (same as sine’s proof): (cos h−1)/h → 0, sin h/h → 1. ∎

The proofs for sine and cosine are mirror images — the only difference is the angle-sum formula’s sign, which propagates straight into the answer.

How to use it

The procedure:

  1. Plain cos x → −sin x. The minus is part of the rule — write it first, before anything else.
  2. Something inside? Chain rule: d/dx [cos(u)] = −sin(u)·u′. Example: d/dx [cos(4x)] = −4 sin(4x).
  3. Constants ride along: d/dx [−5 cos x] = −5·(−sin x) = 5 sin x — minus times minus.
  4. Double-check with the graph: cosine falls first, so near 0 its derivative is negative — if your answer is positive near 0, the minus got lost.

Judgment calls

cos²x = (cos x)² needs the chain: 2 cos x·(−sin x) = −sin 2x. Don’t “cancel” minuses blindly: d/dx [−cos x] = +sin x, but d/dx [cos(−x)] = −sin(−x)·(−1) = −sin x — track each minus to its source.

Common mistake: writing d/dx [cos x] = sin x (dropping the minus). Graph check at π/2: cosine plunges (slope −1), but sin(π/2) = +1. The picture convicts the missing minus instantly.

Worked examples

Four problems, easiest first. The minus sign is the main character.

Example 1 — d/dx [−5 cos x] (minus times minus)

  1. Pull out −5: = −5·d/dx [cos x].
  2. Cosine rule: d/dx [cos x] = −sin x.
  3. Multiply: (−5)(−sin x) = 5 sin x. (Why positive? Two minuses make a plus.)
Common mistake: answering −5 sin x — keeping the −5’s sign but forgetting the rule’s own minus. There are two minuses; both count.
Your turn: d/dx [6 cos x]

Answer: −6 sin x

Pull out the 6: 6·d/dx [cos x] = 6·(−sin x) = −6 sin x. The rule’s minus is non-negotiable.

Example 2 — d/dx [cos(4x)] (chain)

  1. Layers: outer cos u, inner u = 4x.
  2. Outside: −sin(4x). Inside’s derivative: 4.
  3. Multiply: = −4 sin(4x).
Common mistake: 4 sin(4x) — chain factor kept, rule’s minus dropped. Write the minus first, the moment you see cosine.
Your turn: d/dx [cos(2x)] (chain)

Answer: −2 sin(2x)

Layers: outer cos u, inner u = 2x. Outside: −sin(2x). Inside’s derivative: 2. Multiply: −2 sin(2x).

Example 3 — d/dx [x cos x] (product)

  1. Structure: multiplied — product rule: f = x, g = cos x.
  2. Derivatives: f′ = 1, g′ = −sin x.
  3. Assemble f′g + fg′: = 1·cos x + x·(−sin x) = cos x − x sin x.
Common mistake: cos x + x sin x — the product rule’s plus can’t rescue a dropped minus inside g′. Parenthesize (−sin x) when substituting.
Your turn: d/dx [x² cos x] (product)

Answer: 2x cos x − x² sin x

Product rule: f = x², g = cos x. f′ = 2x, g′ = −sin x. Assemble: 2x cos x + x²(−sin x) = 2x cos x − x² sin x.

Example 4 — d/dx [cos²x] (power of cosine)

  1. Rewrite: (cos x)² — outer u², inner cos x.
  2. Outside: 2 cos x. Inside’s derivative: −sin x.
  3. Multiply: = −2 sin x cos x (= −sin 2x).
Common mistake: 2 sin x cos x (positive) — that’s the derivative of sin²x. The two “squared trig” derivatives differ by exactly one minus; label which you’re doing.
Your turn: d/dx [cos³x] (power of cosine)

Answer: −3 cos²x sin x

Rewrite: (cos x)³ — outer u³, inner cos x. Outside: 3 cos²x. Inside’s derivative: −sin x. Multiply: −3 cos²x sin x.

Memorization tips

  • Cosine is the MINUS one: d/dx [cos] = −sin. Write the minus the instant you see cosine — before the chain factor, before everything.
  • Graph check at π/2: cosine plunges there (slope −1). If your formula gives +1 at π/2, the minus is missing. Five seconds, conclusive.
  • One chant for both: “sine to cosine, cosine to negative sine.” Stress the word “negative” — it’s the only asymmetry.
  • Parenthesize the minus: in products and chains, substitute (−sin x) with brackets. The #1 error is a minus lost during substitution.
  • Two minuses make a plus: d/dx [−5 cos x] = +5 sin x. Count minuses like factors — because they are.
  • sin² vs cos²: their derivatives are 2 sin x cos x and −2 sin x cos x — identical except the sign. Label the base before differentiating.

Final challenge

Five mixed questions — the minus sign under pressure. Score 5/5 and it’s 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 derivative of cos x?

d/dx[cos x] = −sin x — with a minus sign. Cosine starts at its peak and falls, so its early slopes are negative; the limit proof traces the minus to the angle-sum formula cos(x+h) = cos x·cos h − sin x·sin h.

Why does cosine’s derivative have a minus but sine’s doesn’t?

Look at x = 0: sine rises (positive slope → +cos) while cosine is at its peak turning downward. The asymmetry is in the waves’ starting positions — sine starts climbing, cosine starts falling.

How do I remember which trig derivative has the minus?

Chant “sine to cosine, cosine to negative sine” — the word “negative” only appears once. Or graph-check at π/2: cosine plunges (slope −1), so its derivative must be negative there.

What is d/dx[cos(4x)]?

−4sin(4x). Chain rule: differentiate the outside (−sin, keeping 4x), then multiply by the inside’s derivative 4. Write the rule’s minus first, then the chain factor.

What is d/dx[−5cos x]?

5sin x. The constant −5 rides along: (−5)·(−sin x) = +5sin x — minus times minus makes a plus.

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