Derivatives: rules and functions

Derivatives of sin x and cos x

The derivative of sin⁡x\sin x is cos⁡x\cos x, and the derivative of cos⁡x\cos x is −sin⁡x-\sin x.

(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x\begin{gathered} (\sin x)'=\cos x, \\[4pt] (\cos x)'=-\sin x \end{gathered}

Differentiating sin⁡\sin gives cos⁡\cos, and differentiating cos⁡\cos gives −sin⁡-\sin. The minus sign appears only when you differentiate cos⁡\cos; dropping it is the most common mistake.

sin⁡x → cos⁡x → −sin⁡x → −cos⁡x → sin⁡x\sin x\ \to\ \cos x\ \to\ -\sin x\ \to\ -\cos x\ \to\ \sin x

Four derivatives bring you back to the start. Both formulas hold only when angles are measured in radians.

xyy = sin x−ππxyy = cos x−ππ
The slope of sin is cos: where sin peaks (slope 0), cos is 0.Drag the dot

Examples

  1. 01Differentiateddxsin⁡x\displaystyle \frac{d}{dx} \sin xAnswerHide
    Answercos⁡x\displaystyle \cos x
  2. 02Differentiateddxcos⁡x\displaystyle \frac{d}{dx} \cos xAnswerHide
    Answer−sin⁡x\displaystyle -\sin x
  3. 03Differentiateddxsin⁡2x\displaystyle \frac{d}{dx} \sin 2xAnswerHide
    Answer2cos⁡2x\displaystyle 2\cos 2x
  4. 04Differentiateddxcos⁡3x\displaystyle \frac{d}{dx} \cos 3xAnswerHide
    Answer−3sin⁡3x\displaystyle -3\sin 3x
  5. 05Differentiateddxsin⁡2x\displaystyle \frac{d}{dx} \sin^{2} xAnswerHide
    Answer2sin⁡xcos⁡x\displaystyle 2\sin x\cos x
  6. 06Differentiateddx(x+cos⁡x)\displaystyle \frac{d}{dx} (x + \cos x)AnswerHide
    Answer1−sin⁡x\displaystyle 1 - \sin x

FAQ

Q1Which one gets the minus sign, sin or cos?Show answerHide
A
The derivative of cos: (cos⁡x)′=−sin⁡x(\cos x)'=-\sin x. There is no minus sign in (sin⁡x)′=cos⁡x(\sin x)'=\cos x.

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Trigonometric functions and radiansChain rule
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Derivatives of sin x and cos x
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Derivative of tan xIntegrals of trigonometric functions

Published 2026-09-03 · Updated 2026-09-05