Derivative of sin(2x)
sin(2x) is the canonical first chain-rule problem: one outer function, one inner function, one multiplication.
Step-by-step solution
The argument of sine is not plain x but the inner function u = 2x. Composite function → chain rule.
d/du[sin(u)] = cos(u), so the outer derivative is cos(2x) with the inside left intact.
d/dx[2x] = 2, giving f′(x) = 2 cos(2x).
Why it works
The factor of 2 is frequency talking. sin(2x) completes its wave twice as fast as sin(x), so at corresponding points it must be climbing and diving twice as steeply — the amplitude of the derivative doubles even though the amplitude of the function doesn't change. In wave language: same height, double the frequency, double the maximum slope.
You can double-check the result with the identity sin(2x) = 2 sin(x)cos(x): differentiate the right side with the product rule and you get 2[cos²(x) − sin²(x)] = 2 cos(2x), agreeing with the chain rule answer. When two different methods land on the same expression, you can be very confident in both.
Common mistakes
- Answering cos(2x) with no factor of 2 — the chain rule's multiplication step skipped entirely.
- Changing the argument: writing 2 cos(x). The inside function survives untouched; only a factor pops out front.
- Applying a "double angle" reflex and answering 2 cos(x)sin(x) or similar identity fragments — identities are for rewriting, not differentiating.
Practice problems
Differentiate sin(5x)
Answer: 5 cos(5x).
Differentiate 3 sin(2x)
Answer: 6 cos(2x).
Differentiate sin(2x) + sin(x)
Answer: 2 cos(2x) + cos(x).