Derivative of tan x

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

Writing tan x = sin x / cos x and applying the quotient rule gives (cos²x + sin²x)/cos²x = 1/cos²x = sec²x. The slope is never negative, and it shoots up near the asymptotes at x = ±π/2.

Derivation

Trig rule
$\frac{d}{dx}\left[\tan\left(x\right)\right] = \sec^{2}\left(x\right)$

f(x) = tan x
f′(x)

Drag the graph · at x = 0.42 the slope is 1.1994, the height of the dashed curve.

Open in the derivative calculator

Questions students ask

What is the derivative of tan x?

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

How do you find the derivative of tan x?

Writing tan x = sin x / cos x and applying the quotient rule gives (cos²x + sin²x)/cos²x = 1/cos²x = sec²x. The slope is never negative, and it shoots up near the asymptotes at x = ±π/2.

What is the integral of tan x?

See the integral of tan x page for the antiderivative with steps.