AP Calculus AB and BC

Derivative of tan 2x: Answer, Proof, and Mistakes

Q: What is the derivative of \tan 2x?

It is 2\sec^2 2x, equivalently \frac{2}{\cos^2 2x}. The 2 comes from the chain rule, the derivative of the inside 2x.

Q: What is the derivative of \tan(kx)?

It is k\sec^2(kx) for any constant k, by the same chain rule step with inner derivative k.

The derivative of tan 2x is 2 sec^2(2x). In prime notation, if f(x) = tan 2x then f'(x) = 2 sec^2(2x). The chain rule differentiates the outer tangent into sec^2 of the inside, then multiplies by the inner derivative 2, since the derivative of tan u is sec^2(u) times u'.

\frac{d}{dx}\left[\tan 2x\right] = 2\sec^2 2x

How to differentiate tan 2x

The inner function is $2x$ and the outer is tangent. Differentiate the outer with the standard rule $\frac{d}{dx}\tan u = \sec^2 u$ , hold the inside unchanged, then multiply by the derivative of the inside.

\frac{d}{dx}\tan(2x) = \sec^2(2x)\cdot 2 = 2\sec^2 2x

The general linear-inner version follows the same way and is worth carrying as one fact.

\frac{d}{dx}\tan(kx) = k\sec^2(kx)

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

The chain rule is Topic 3.1 in Unit 3, on both AB and BC, and a trig function with a linear inside is one of its first standard examples. You are expected to recall $\frac{d}{dx}\tan x = \sec^2 x$ and combine it with the inner factor of $2$ .

Because $\sec^2 2x = \frac{1}{\cos^2 2x}$ is positive wherever it is defined, $\tan 2x$ is always increasing between its asymptotes, which shows up in sign-of-the-derivative and increasing/decreasing questions.

Common mistakes with the derivative of tan 2x

Answering $\sec^2 2x$ and dropping the inner factor of $2$ . The chain rule is what puts it there.
Writing $2\sec^2 x$ instead of $2\sec^2 2x$ , differentiating the inside but forgetting to keep it inside the secant.
Confusing it with the product $\frac{d}{dx}(2\tan x) = 2\sec^2 x$ , which is a different function than $\tan 2x$ .

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 $\tan 2x$ ?

It is $2\sec^2 2x$ , equivalently $\frac{2}{\cos^2 2x}$ . The $2$ comes from the chain rule, the derivative of the inside $2x$ .

Why is there a factor of 2 in the answer?

The chain rule multiplies by the derivative of the inner function $2x$ , which is $2$ . Without that step you would only have $\sec^2 2x$ .

What is the derivative of $\tan(kx)$ ?

It is $k\sec^2(kx)$ for any constant $k$ , by the same chain rule step with inner derivative $k$ .