AP Calculus AB and BC

Derivative of cos 4x: Answer and Chain Rule

Q: What is the second derivative?

-16\cos 4x, which is -16 times the original function.

The derivative of cos 4x is negative 4 sin 4x. Cosine differentiates to negative sine with the argument unchanged, and the chain rule separately multiplies by 4 for the inner function.

\frac{d}{dx}\left[\cos 4x\right] = -4\sin 4x

Two independent contributions

\frac{d}{dx}\cos(4x) = -\sin(4x)\cdot 4 = -4\sin 4x

The minus is from cosine, the $4$ is from the chain rule, and they come from different steps. Checking at $x = 0$ confirms it: the function is at a maximum there and $-4\sin 0 = 0$ .

The second derivative is proportional

\frac{d^{2}}{dx^{2}}\cos 4x = -16\cos 4x

That is $-16$ times the original function, which is why cosines of this form solve simple harmonic motion equations.

Common mistakes

Answering $4\sin 4x$ with no minus sign.
Answering $-\sin 4x$ , dropping the chain rule factor.
Answering $-4\sin x$ , moving the coefficient out of 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 4x?

It is $-4\sin 4x$ .

What is the second derivative?

$-16\cos 4x$ , which is $-16$ times the original function.