AP Calculus AB and BC

Derivative of cos^2 x: Answer, Chain Rule, Mistakes

The derivative of cos squared x with respect to x is negative 2 sin x cos x, which equals negative sin 2x. The chain rule gives twice cosine times the derivative of cosine, and that inner derivative is where the minus sign comes from.

ddx[cos⁡2x]=−2sin⁡xcos⁡x\frac{d}{dx}\left[\cos^{2} x\right] = -2\sin x\cos x

How to differentiate cos^2 x

Treat it as (cos⁡x)2\left(\cos x\right)^{2}. The power rule handles the outer square, and the chain rule multiplies by the derivative of the inside.

ddx(cos⁡x)2=2cos⁡x⋅(−sin⁡x)\frac{d}{dx}\left(\cos x\right)^{2} = 2\cos x\cdot\left(-\sin x\right)
ddxcos⁡2x=−2sin⁡xcos⁡x=−sin⁡2x\frac{d}{dx}\cos^{2} x = -2\sin x\cos x = -\sin 2x

Where the minus sign comes from

Only one place: the derivative of the inner cosine. The outer power rule contributes the 22 and nothing else, so a lost minus sign is always the inner step.

Check at x=0x = 0. There cos⁡2x\cos^{2}x is at its maximum of 11, so the derivative should be 00, and −2sin⁡0cos⁡0=0-2\sin 0\cos 0 = 0 agrees.

Where the derivative of cos^2 x shows up on the AP exam

Topic 3.1 on both exams. It pairs naturally with the integral of cos⁡2x\cos^{2}x, which needs the power reducing identity rather than the chain rule, and the contrast is a good check on whether a student knows which direction they are working in.

Common mistakes with the derivative of cos^2 x

  • Answering 2cos⁡x2\cos x, dropping the inner derivative entirely.
  • Answering 2sin⁡xcos⁡x2\sin x\cos x with no minus sign, which is the derivative of sin⁡2x\sin^{2}x instead.
  • Answering −sin⁡2x-\sin^{2} x, which mixes the outer and inner steps together.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of cos^2 x?

It is −2sin⁡xcos⁡x-2\sin x\cos x, equivalently −sin⁡2x-\sin 2x.

Why is it negative?

The chain rule multiplies by the derivative of the inner cosine, which is −sin⁡x-\sin x.

What is the integral of cos^2 x?

It is x2+sin⁡2x4+C\frac{x}{2} + \frac{\sin 2x}{4} + C, and it needs the power reducing identity, not the chain rule.

How does it relate to the derivative of sin^2 x?

It is the exact negative. The two derivatives sum to zero because sin⁡2x+cos⁡2x\sin^{2}x + \cos^{2}x is the constant 11.