AP Calculus AB and BC

Derivative of cos^3 x: Answer, Proof, and Mistakes

The derivative of cos^3 x is -3 sin(x) cos^2(x). In prime notation, if f(x) = cos^3 x then f'(x) = -3 sin(x) cos^2(x). Read cos^3 x as (cos x)^3, differentiate the outer cube into 3(cos x)^2, then multiply by the inner derivative -sin x, which supplies the minus sign.

ddx[cos⁡3x]=−3sin⁡xcos⁡2x\frac{d}{dx}\left[\cos^3 x\right] = -3\sin x\cos^2 x

How to differentiate cos^3 x

The notation cos⁡3x\cos^3 x means (cos⁡x)3(\cos x)^3. The outer function is the cube and the inner is cos⁡x\cos x, so use ddxu3=3u2 u′\frac{d}{dx}u^3 = 3u^2\,u' with u=cos⁡xu = \cos x and u′=−sin⁡xu' = -\sin x.

ddx(cos⁡x)3=3(cos⁡x)2(−sin⁡x)=−3sin⁡xcos⁡2x\frac{d}{dx}(\cos x)^3 = 3(\cos x)^2(-\sin x) = -3\sin x\cos^2 x

The minus sign comes entirely from the inner derivative ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin x. It is the single easiest part of this problem to lose.

Reading the notation, and where it appears

cos⁡3x\cos^3 x is a power of a trig function, differentiated by the chain rule (Topic 3.1). It is not cos⁡(x3)\cos(x^3), whose derivative is −3x2sin⁡(x3)-3x^2\sin(x^3). It is also the exact structural twin of sin⁡3x\sin^3 x, differing only by the sign that the inner derivative carries.

On BC the same cos⁡2xsin⁡x\cos^2 x\sin x pattern signals a u=cos⁡xu = \cos x substitution when you integrate, so recognizing it both ways is useful.

Common mistakes with the derivative of cos^3 x

  • Answering 3cos⁡2xsin⁡x3\cos^2 x\sin x with no minus sign. The inner derivative of cosine is −sin⁡x-\sin x, so the result is negative.
  • Answering 3cos⁡2x3\cos^2 x and dropping the inner derivative entirely.
  • Reading cos⁡3x\cos^3 x as cos⁡(x3)\cos(x^3) and answering −3x2sin⁡(x3)-3x^2\sin(x^3), a different function.

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⁡3x\cos^3 x?

It is −3sin⁡xcos⁡2x-3\sin x\cos^2 x. The minus sign comes from the inner derivative of cos⁡x\cos x, which is −sin⁡x-\sin x.

Why is there a minus sign?

The chain rule multiplies by the inner derivative ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin x, and that negative sign carries into the final answer.

How is cos⁡3x\cos^3 x related to sin⁡3x\sin^3 x?

They have the same chain rule structure. The derivative of sin⁡3x\sin^3 x is 3sin⁡2xcos⁡x3\sin^2 x\cos x; the derivative of cos⁡3x\cos^3 x is its sign-flipped cousin −3sin⁡xcos⁡2x-3\sin x\cos^2 x.