weirdfacts

With x measured in radians, the derivative of sin⁡x\sin x is cos⁡x\cos x and that of cos⁡x\cos x is −sin⁡x-\sin x. A frequency multiplier inside the angle also multiplies the derivative. The minus sign for cosine matters.

Products such as xsin⁡xx\sin x need the product rule, and a quotient needs both numerator terms in the correct order. Simplifying a derivative can reveal stationary inputs or useful factors. Check a numerical gradient as a reasonableness test, not as a replacement for exact working.

INTERACTIVE MODEL

Try it. Watch it change.

Function sin(x), tangent and derivative-2.5-1.88-1.25-0.6300.631.251.882.5-5-1.751.54.758xy
f(x)f(x) = sin⁡(x)\sin(x)tangentf′(x)f'(x)
f′(1) = 0.5403 · f″(1) = -0.8415

The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.

Explore: Select sin⁡(x)\sin(x). Compare the tangent slope with cos⁡(x)\cos(x), especially at a peak and a zero.

Sine derivative
ddxsin⁡x=cos⁡x\frac{\mathrm d}{\mathrm dx}\sin x=\cos x
Cosine derivative
ddxcos⁡x=−sin⁡x\frac{\mathrm d}{\mathrm dx}\cos x=-\sin x
WORKED EXAMPLE

Differentiate f(x)f(x) = xsin⁡(2x)x\sin(2x).

  1. Differentiate each factor in turn.
  2. f′(x)f'(x) = sin⁡(2x)\sin(2x) + 2xcos⁡(2x)2x\cos(2x).
Assumed knowledge

Unit circle, radians and differentiation rules.

Learning checkpoints & sourceYOUR LEARNING CHECKPOINT
  • Differentiate sine and cosine in radians.
  • Combine chain, product and quotient rules.
QCAA Mathematical Methods 2025 v1.3 · p. 24 ↗
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