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9.1PureStretch

Differentiating sin x and cos x

Using the small angle approximations and the limit definition, we can differentiate sin x and cos x from first principles. The results are clean and must be memorised — but only hold when x is in radians.

26 min Differentiation
Edexcel A level Maths: 9.1 First Principles, Differentiating Sin(x) and Cos(x)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Differentiate sin x and cos x
  • Understand the first-principles derivation
  • Apply the results to gradients
  • Know they require radians
1

The results

From first principles (using sin⁡θ≈θ\sin\theta \approx \theta for small θ\theta): 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.

ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x
Derivative of sine.
ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x
Derivative of cosine (note the minus).
2

Why radians?

The first-principles limit lim⁡h→0sin⁡hh=1\lim_{h\to 0}\tfrac{\sin h}{h} = 1 only holds when hh is in radians, so these derivatives are only valid in radians.

Example 1Find the gradient of y=sin⁡xy = \sin x at x=0x = 0.
1dydx=cos⁡x\tfrac{dy}{dx} = \cos x.
2At x=0x=0: cos⁡0=1\cos 0 = 1.
Answer
Gradient =1= 1

Tip — Remember the minus: d/dx(cos x) = −sin x.

Formula recap

ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x
Sine.
ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x
Cosine.

Common mistakes to avoid

Differentiating cos x to +sin x.
It is −sin x.
Using these derivatives in degrees.
They only hold for x in radians.

Key takeaways

  • d/dx(sin x) = cos x.
  • d/dx(cos x) = −sin x.
  • Valid only in radians (from the sin h / h limit).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ddx(sin⁡x)\frac{d}{dx}(\sin x) equals:

    • cos⁡x\cos x
    • −cos⁡x-\cos x
    • sin⁡x\sin x
    • −sin⁡x-\sin x
    Show answer and explanation

    Answer: cos⁡x\cos x

    Standard result.

  2. ddx(cos⁡x)\frac{d}{dx}(\cos x) equals:

    • −sin⁡x-\sin x
    • sin⁡x\sin x
    • cos⁡x\cos x
    • −cos⁡x-\cos x
    Show answer and explanation

    Answer: −sin⁡x-\sin x

    Note the minus.

  3. These derivatives require x in:

    • radians
    • degrees
    • gradians
    • any unit
    Show answer and explanation

    Answer: radians

    From the limit sin h / h → 1.

  4. The gradient of y=sin⁡xy = \sin x at x=0x = 0 is:

    • 11
    • 00
    • −1-1
    • π\pi
    Show answer and explanation

    Answer: 11

    cos 0 = 1.

  5. The key limit used is lim⁡h→0sin⁡hh=\lim_{h\to0}\tfrac{\sin h}{h} =

    • 11
    • 00
    • ∞\infty
    • hh
    Show answer and explanation

    Answer: 11

    Small angle.

Next step

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