Trigonometric derivatives

Derivative of sin(x): Why It Is cos(x)

The derivative of sine measures instantaneous vertical speed around the unit circle. Radian measure aligns angle with arc length, so no conversion constant appears.

Direct answer

ddx[sin⁡x]=cos⁡x\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin x\right] = \cos x

The standard formula assumes radians; sin(g(x)) also requires multiplication by g′(x).

When to use derivative of sin(x)

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Use the sine rule for sin(g(x)), then inspect g(x) for a chain-rule factor.

Ask first 2

Which condition can change the answer?

What to do

The standard formula assumes radians; sin(g(x)) also requires multiplication by g′(x).

Ask first 3

How should the result be checked?

What to do

Confirm the original angle remains inside cosine and every inner rate is present.

Difference-quotient derivation

Expand sin(x+h) with the angle-addition identity inside the difference quotient.

Separate sin(x)[cos(h)−1]/h from cos(x)sin(h)/h.

The first standard limit is zero and the second is one, leaving cos(x).

For sin(g(x)), keep g(x) inside cosine and multiply by g′(x).

1
sin⁡(x+h)=sin⁡x cos⁡h+cos⁡x sin⁡h\sin\left(x + h\right) = \sin x\,\cos h + \cos x\,\sin h
2
lim sin(h)/h=1
3
[sin(g(x))]'=cos(g(x))g'(x)

Worked examples

ddx[sin⁡x]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin x\right]
cos⁡x\cos x

Direct sine rule in radians.

ddx[sin⁡(3 x)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin\left(3\,x\right)\right]
3 cos⁡(3 x)3\,\cos\left(3\,x\right)

Multiply by the inner derivative 3.

ddx[sin⁡(x2)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin\left(x^{2}\right)\right]
2x cos(x²)

Preserve x² inside cosine and multiply by 2x.

Common mistakes

  • • Writing −sin(x), which belongs to cosine.
  • • Using degrees without π/180.
  • • Dropping the inner derivative.
  • • Replacing g(x) by g′(x) inside cosine.

Error clinic: locate the first invalid step

Incorrect attempt

[sin(x²)]'=cos(x²)

Why it fails

The inner rate 2x is missing.

Correction

[sin(x²)]'=2x cos(x²)

Incorrect attempt

[sin(x)]'=−sin(x)

Why it fails

This confuses the sine and cosine derivative cycle.

Correction

[sin(x)]'=cos(x)

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

ddx[sin⁡(3 x)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin\left(3\,x\right)\right]

Check before solving

Multiply by the inner derivative 3.

Reveal the reference answer

Reference answer

3 cos⁡(3 x)3\,\cos\left(3\,x\right)
ddx[sin⁡(x2)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin\left(x^{2}\right)\right]

Check before solving

Preserve x² inside cosine and multiply by 2x.

Reveal the reference answer

Reference answer

2x cos(x²)

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Derivatives:sin⁡x\sin xDerivatives:sin⁡(x2)\sin\left(x^{2}\right)
Open the calculator without a preset: Derivatives

Frequently asked questions

Why must x be in radians?+

The standard limits used in the proof take their unit values in radian measure.

What is the second derivative?+

Differentiate cos(x) to obtain −sin(x).

Does sin(x²) differentiate to cos(x²)?+

No. The chain rule adds 2x.

Conclusion

In radians, sine differentiates to cosine. For a composite angle, preserve the angle and multiply by its derivative.

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