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

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.

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

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 $\sec 2x$.