AP Calculus AB and BC

Derivative of tan^2 x: Answer, Chain Rule, Mistakes

Q: Why is it the same as the derivative of sec^2 x?

Because \tan^{2}x = \sec^{2}x - 1, and functions differing by a constant have identical derivatives.

Q: What is the integral of tan^2 x?

It is \tan x - x + C, using the identity to rewrite the integrand as \sec^{2}x - 1.

Q: Where is tan^2 x not differentiable?

Wherever tangent is undefined, at odd multiples of \frac{\pi}{2}.

The derivative of tan squared x with respect to x is 2 tan x times sec squared x. The outer function is the square and the inner is tangent, whose derivative is sec squared x. The same answer also equals the derivative of sec squared x, because the two functions differ by a constant.

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

How to differentiate tan^2 x

Apply the power rule to the outer square, then multiply by the derivative of tangent.

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

Why it equals the derivative of sec^2 x

The Pythagorean identity says $\tan^{2}x = \sec^{2}x - 1$ . Two functions differing by a constant have the same derivative, so differentiating $\sec^{2}x$ must give the same answer.

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

The two routes agree, which is a useful check and a good example of why a constant difference is invisible to differentiation.

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

Topic 3.1 on both exams. It is also the reason $\int \tan^{2}x\,dx = \tan x - x + C$ : the identity converts the integrand into $\sec^{2}x - 1$ , which is two basic antiderivatives.

Common mistakes with the derivative of tan^2 x

Answering $2\tan x$ and dropping the inner derivative $\sec^{2}x$ .
Answering $\sec^{4}x$ , which squares the derivative instead of applying the chain rule.
Assuming the two forms $2\tan x\sec^{2}x$ and $2\sec^{2}x\tan x$ differ. They are the same product written in a different order.

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^2 x?

It is $2\tan x\sec^{2} x$ .

Why is it the same as the derivative of sec^2 x?

Because $\tan^{2}x = \sec^{2}x - 1$ , and functions differing by a constant have identical derivatives.

What is the integral of tan^2 x?

It is $\tan x - x + C$ , using the identity to rewrite the integrand as $\sec^{2}x - 1$ .

Where is tan^2 x not differentiable?

Wherever tangent is undefined, at odd multiples of $\frac{\pi}{2}$ .