Calculus I › Differentiation rules › full formula sheet
The derivative of sin x
The sine wave’s slope pattern is the cosine wave — no minus sign, and the limit proof shows exactly why.
Notation: x is in radians (calculus always uses radians). d/dx[sin(u)] = cos(u)·u′ when there’s something inside.
Before this lesson: Definition of the derivative · Key trig limit
Where it comes from
Picture the sine wave. At x = 0 it crosses the axis heading up at 45° — slope +1. Which candidate matches? cos(0) = 1, while −cos(0) = −1. So the derivative can’t carry a minus sign — the graph kills −cos instantly. The slope pattern of sine (1 at 0, 0 at π/2, −1 at π, …) traces out exactly the cosine wave.
The intuition: sine and cosine are the same wave shifted. Where sine climbs fastest, cosine peaks; where sine peaks, cosine crosses zero. The derivative “shifts” the wave by a quarter period — and shifting sine by a quarter period gives cosine:
Before reading on: sine climbs through x = 0 at 45°, so its slope there is +1. Which candidate — cos x or −cos x — matches, and what does that rule out?
This is the plus trig derivative — the minus sign belongs to cosine’s derivative, not sine’s. Keep them straight and half the chapter’s sign errors vanish.
Derivation
Apply the definition to f(x) = sin x, using the angle-sum formula sin(x+h) = sin x cos h + cos x sin h and two famous limits:
Where do those limits come from? lim sin h/h = 1 is proved with the squeeze theorem on a unit-circle wedge (it’s on the Limits sheet as the key trig limit); lim (cos h−1)/h = 0 follows from it with a half-angle identity. Everything rests on geometry, not magic.
How to use it
The procedure:
- Plain sin x → cos x. No minus sign. Ever.
- Something inside? Chain rule: d/dx [sin(u)] = cos(u)·u′. Example: d/dx [sin(5x)] = 5 cos(5x).
- Constant multiples ride along: d/dx [3 sin x] = 3 cos x.
- Products need the product rule: x² sin x is multiplied, not composed — don’t chain it.
Judgment calls
sin²x means (sin x)² — a composition, so chain: 2 sin x cos x. Radians only: this formula assumes x in radians; in degrees an extra π/180 factor appears (calculus always uses radians, so you’ll rarely meet this).
Worked examples
Four problems, easiest first. Watch where the chain rule fires.
Example 1 — d/dx [3 sin x]
- Pull out the 3 (constant multiple): = 3·d/dx [sin x].
- Sine rule: = 3 cos x = 3 cos x.
Your turn: d/dx [−4 sin x]
Answer: −4 cos x
Pull out the −4: −4·d/dx [sin x] = −4 cos x. No minus appears — sine is the plus one.
Example 2 — d/dx [sin(2x)] (chain)
- Layers: outer sin u, inner u = 2x.
- Outside: cos(2x). Inside’s derivative: 2.
- Multiply: = 2 cos(2x).
Your turn: d/dx [sin(7x)] (chain)
Answer: 7 cos(7x)
Layers: outer sin u, inner u = 7x. Outside: cos(7x). Inside’s derivative: 7. Multiply: 7 cos(7x).
Example 3 — d/dx [x² sin x] (product)
- Structure: multiplied, not composed — product rule: f = x², g = sin x.
- Derivatives: f′ = 2x, g′ = cos x.
- Assemble f′g + fg′: = 2x sin x + x² cos x.
Your turn: d/dx [x sin x] (product)
Answer: sin x + x cos x
Multiplied, not composed — product rule: f = x, g = sin x. f′ = 1, g′ = cos x. Assemble: 1·sin x + x·cos x.
Example 4 — d/dx [sin²x] (power of sine)
- Rewrite: (sin x)² — outer u², inner sin x.
- Outside: 2 sin x. Inside’s derivative: cos x.
- Multiply: = 2 sin x cos x (= sin 2x by the double-angle identity).
Your turn: d/dx [sin³x] (power of sine)
Answer: 3 sin²x cos x
Rewrite: (sin x)³ — outer u³, inner sin x. Outside: 3 sin²x. Inside’s derivative: cos x. Multiply: 3 sin²x cos x.
Memorization tips
- Sine is the PLUS one: d/dx [sin] = +cos. The minus belongs to cosine. Mix them and every trig derivative flips.
- Graph check in 2 seconds: sine rises at 0 → slope positive → cos(0) = +1 confirms no minus. Run it whenever you doubt.
- Quarter-period shift: differentiating shifts the wave left by π/2; sine shifted is cosine. The picture is the formula.
- Inside? Chain: sin(2x), sin(x²), sin²x all need ×u′. Plain sin x is the only one that doesn’t.
- Radians, always: the proof’s key limit sin h/h → 1 holds for radian h. In calculus, trig means radians — no exceptions.
- Pair it with cosine: learn both as one chant — “sine to cosine, cosine to negative sine.” Two facts, one breath.
Final challenge
Five mixed questions — the minus-sign trap, chain versions, and products. Score 5/5 and it’s 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 derivative of sin x?
d/dx[sin x] = cos x — with no minus sign. The slope pattern of the sine wave (1 at 0, 0 at π/2, −1 at π) traces out exactly the cosine wave.
Why is there no minus sign?
The graph forbids it: sine rises through x = 0, so its slope there is positive (+1), matching cos(0) = 1. A minus would give −1, contradicting the picture. (The minus belongs to cosine’s derivative.)
How is d/dx[sin x] = cos x proved?
From the limit definition with sin(x+h) = sin x·cos h + cos x·sin h. Regrouping exposes the two key limits lim(sin h/h) = 1 and lim((cos h−1)/h) = 0, leaving sin x·0 + cos x·1 = cos x.
What is d/dx[sin(2x)]?
2cos(2x) — the chain rule. Differentiate the outside (cos, keeping 2x), then multiply by the inside’s derivative 2. Forgetting the ×2 is the classic error.
Does x have to be in radians?
Yes. The proof depends on lim(sin h/h) = 1, which holds only for radian measure. Calculus always uses radians for trig functions — in degrees an extra π/180 factor would appear.
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