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.
Domain and tangent line
Sin(2x) is differentiable for every real x, with angles in radians. At x = 0 the point is (0, 0) and the slope is 2, so the tangent line is y = 2x. At x = π/4 its value is 1 and its slope is zero.
More generally, sin(kx) has derivative k cos(kx) for any real constant k. Its maximum absolute slope is |k|. When k = 0 the function is constant and its derivative is zero.
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).