AP Calculus AB and BC

Derivative of tan x / x: Quotient Rule

The derivative of tan x over x is x times secant squared x, minus tan x, all over x squared. The quotient rule gives it in one step once you know tangent differentiates to secant squared.

ddx[tan⁡xx]=xsec⁡2x−tan⁡xx2\frac{d}{dx}\left[\frac{\tan x}{x}\right] = \frac{x\sec^{2}x - \tan x}{x^{2}}

The quotient rule

xsec⁡2x−tan⁡x⋅1x2\frac{x\sec^{2}x - \tan x\cdot 1}{x^{2}}

The expression is undefined at x=0x = 0 and wherever tangent is, so the derivative inherits both sets of exclusions.

The removable hole at the origin

Although undefined at 00, the function has limit 11 there, so the discontinuity is removable. That is the standard limit lim⁡x→0tan⁡xx=1\lim_{x \to 0}\frac{\tan x}{x} = 1, and it is why the graph looks continuous through the origin.

Common mistakes

  • Using sec⁡xtan⁡x\sec x\tan x, which is the derivative of secant, not tangent.
  • Assuming the origin is an asymptote. The limit exists and equals 11.

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 x / x?

It is xsec⁡2x−tan⁡xx2\frac{x\sec^{2}x - \tan x}{x^{2}}.

What kind of discontinuity is at x = 0?

Removable. The limit there is 11.