AP Calculus AB and BC

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

Q: What is the second derivative of cos 2x?

It is -4\cos 2x, which is -4 times the original function.

The derivative of cos 2x with respect to x is negative 2 sin 2x. The chain rule differentiates the outer cosine to negative sin 2x and then multiplies by 2, the derivative of the inner function. In general cos kx differentiates to negative k sin kx.

\frac{d}{dx}\left[\cos 2x\right] = -2\sin 2x

How to differentiate cos 2x

Two things happen at once: cosine differentiates to negative sine, and the chain rule contributes the inner derivative $2$ .

\frac{d}{dx}\cos(2x) = -\sin(2x)\cdot 2 = -2\sin 2x

\frac{d}{dx}\cos(kx) = -k\sin(kx)

Two separate places to lose a sign

The minus comes from differentiating cosine, and it has nothing to do with the chain rule. Students who track only one of the two steps usually keep the $2$ and drop the sign.

A quick sanity check: at $x = 0$ , $\cos 2x$ is at a maximum, so its derivative must be $0$ . Substituting gives $-2\sin 0 = 0$ , which agrees.

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

Topic 3.1 (The Chain Rule) on both exams. It also appears constantly inside simple harmonic motion problems, where position is a cosine and velocity is this derivative.

The second derivative is $-4\cos 2x$ , which is $-4$ times the original function. That relationship is what makes it a solution of a differential equation.

Common mistakes with the derivative of cos 2x

Answering $2\sin 2x$ with no minus sign.
Answering $-\sin 2x$ , dropping the chain rule factor.
Answering $-2\sin x$ , differentiating the inner function into the argument.

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 2x?

It is $-2\sin 2x$ .

Where does the minus sign come from?

From differentiating cosine, which gives negative sine. The chain rule separately contributes the factor of $2$ .

What is the second derivative of cos 2x?

It is $-4\cos 2x$ , which is $-4$ times the original function.

What is the derivative of cos(kx)?

It is $-k\sin(kx)$ .