AP Calculus AB and BC

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

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

It is 2\sec 2x\tan 2x. The 2 is the chain rule factor from the inside 2x.

Q: Why does the answer keep both \sec and \tan?

The derivative of \sec u is \sec u\tan u, so both factors carry through, each with the same argument 2x.

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

It is k\sec(kx)\tan(kx) for any constant k, by the same chain rule step.

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

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

How to differentiate sec 2x

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

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

The general linear-inner form is the same fact with the constant carried through.

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

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

This is a Topic 3.1 chain rule application on AB and BC. The secant derivative $\sec u\tan u$ is one of the six trig derivatives you recall, and the inner $2x$ supplies the factor of $2$ .

It also appears in reverse: $\sec 2x\tan 2x$ is the kind of expression whose antiderivative is $\frac{1}{2}\sec 2x + C$ , so recognizing the pattern both ways saves time.

Common mistakes with the derivative of sec 2x

Answering $\sec 2x\tan 2x$ and dropping the inner factor of $2$ .
Writing $2\sec x\tan x$ , differentiating the inside but forgetting to keep the argument as $2x$ in both the secant and the tangent.
Mixing up the secant derivative with the tangent derivative and writing $2\sec^2 2x$ , which is the derivative of $\tan 2x$ , not $\sec 2x$ .

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 $\sec 2x$ ?

It is $2\sec 2x\tan 2x$ . The $2$ is the chain rule factor from the inside $2x$ .

Why does the answer keep both $\sec$ and $\tan$ ?

The derivative of $\sec u$ is $\sec u\tan u$ , so both factors carry through, each with the same argument $2x$ .

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

It is $k\sec(kx)\tan(kx)$ for any constant $k$ , by the same chain rule step.