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

How to differentiate sin 2x

The inner function is $2x$ and the outer is sine. Differentiate the outer, keep the inner unchanged, then multiply by the inner derivative.

The general rule follows the same way, and it is worth carrying as one fact rather than rederiving each time.

Why the 2 appears

Doubling the input doubles how fast the argument moves, so the output oscillates twice as fast and its slopes are twice as steep. The factor is not bookkeeping, it is the actual rate.

Reversed, the same factor explains the division in the antiderivative: $\int \sin 2x\,dx = -\frac{\cos 2x}{2} + C$.

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

The chain rule is Topic 3.1, on both AB and BC, and a trigonometric function with a linear inner function is the most common first example of it.

Do not confuse it with the product $\frac{d}{dx}\left(2\sin x\right) = 2\cos x$. Those look similar written quickly and are different functions.

Common mistakes with the derivative of sin 2x

• Answering $\cos 2x$ and dropping the chain rule factor. This is the single most common chain rule error.
• Answering $2\cos x$, which differentiates the inner function into the wrong place.
• Using the double angle identity first. Rewriting as $2\sin x\cos x$ and applying the product rule is legal and gives $2\cos 2x$ after simplification, but it is slower and invites algebra errors.