AP Calculus AB and BC

Derivative of cos 3x: Answer, Chain Rule, Mistakes

The derivative of cos 3x with respect to x is -3 sin 3x. The chain rule differentiates the outer cosine to -sin 3x, then multiplies by the derivative of the inner function 3x, which is 3. Two things change together here: the cosine introduces a minus sign and the inner 3x introduces the factor of 3.

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

How to differentiate cos 3x

The outer function is cosine and the inner is 3x3x. Cosine differentiates to negative sine, and the chain rule then multiplies by the inner derivative.

ddxcos⁡(3x)=−sin⁡(3x)⋅3=−3sin⁡3x\frac{d}{dx}\cos(3x) = -\sin(3x)\cdot 3 = -3\sin 3x

The general rule keeps both features, the sign flip and the constant factor.

ddxcos⁡(kx)=−ksin⁡(kx)\frac{d}{dx}\cos(kx) = -k\sin(kx)

Two changes at once: the sign and the factor

Cosine starts at a peak, where the slope is 00 and about to fall, so its derivative leads with a minus sign. Independently, the inner 3x3x contributes a factor of 33 through the chain rule. Both must appear, which is what makes this a frequent two-part error.

Reversed, the sign returns to positive and the factor divides: ∫cos⁡3x dx=13sin⁡3x+C\int \cos 3x\,dx = \frac{1}{3}\sin 3x + C.

Where the derivative of cos 3x shows up on the AP exam

The chain rule is Topic 3.1, on both AB and BC, and Unit 3 is weighted 5 to 10 percent on each. A cosine with a linear inner function tests whether you track the sign and the factor at the same time.

Contrast the constant multiple ddx(3cos⁡x)=−3sin⁡x\frac{d}{dx}(3\cos x) = -3\sin x, where the 33 is a coefficient, with cos⁡3x\cos 3x, where the 33 is inside and enters through the chain rule.

Common mistakes with the derivative of cos 3x

  • Answering −sin⁡3x-\sin 3x and dropping the factor of 33.
  • Answering 3sin⁡3x3\sin 3x and losing the minus sign. Cosine differentiates to negative sine.
  • Answering −3sin⁡x-3\sin x, pulling the 33 out of the argument instead of leaving it inside the sine.

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

Why is the derivative of cos 3x negative?

Cosine differentiates to −sin⁡-\sin, so the minus sign comes from the outer function. The chain rule then attaches the inner derivative 33, giving −3sin⁡3x-3\sin 3x.

What is the derivative of cos(kx)?

ddxcos⁡(kx)=−ksin⁡(kx)\frac{d}{dx}\cos(kx) = -k\sin(kx). The cosine supplies the minus sign and the sine, while the chain rule supplies the factor kk. With k=3k = 3 that is −3sin⁡3x-3\sin 3x.

What is the second derivative of cos 3x?

Differentiate −3sin⁡3x-3\sin 3x to get −9cos⁡3x-9\cos 3x. Each derivative multiplies by 33 and advances cosine, negative sine, negative cosine, and so on.