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derivative of sec x — Definition, Formula & Examples

The derivative of sec x is sec x tan x. This means that when you differentiate the secant function with respect to x, you multiply secant by tangent.

If f(x)=sec⁡xf(x) = \sec x, then f′(x)=sec⁡xtan⁡xf'(x) = \sec x \tan x for all xx in the domain of sec⁡x\sec x (i.e., where cos⁡x≠0\cos x \neq 0). This result follows from rewriting sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x} and applying the quotient rule.

Key Formula

ddx[sec⁡x]=sec⁡xtan⁡x\frac{d}{dx}[\sec x] = \sec x \tan x
Where:
  • xx = The angle, measured in radians

How It Works

Since sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}, you can derive the formula with the quotient rule. Differentiating gives 0⋅cos⁡x−1⋅(−sin⁡x)cos⁡2x=sin⁡xcos⁡2x\frac{0 \cdot \cos x - 1 \cdot (-\sin x)}{\cos^2 x} = \frac{\sin x}{\cos^2 x}. Rewrite this as 1cos⁡x⋅sin⁡xcos⁡x=sec⁡xtan⁡x\frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \sec x \tan x. The result applies wherever cosine is nonzero, so it excludes x=π2+nπx = \frac{\pi}{2} + n\pi for any integer nn.

Worked Example

Problem: Find the derivative of f(x) = 3 sec x.
Apply the constant multiple rule: Pull the constant 3 outside the derivative.
f′(x)=3⋅ddx[sec⁡x]f'(x) = 3 \cdot \frac{d}{dx}[\sec x]
Use the derivative formula: Replace the derivative of sec x with sec x tan x.
f′(x)=3sec⁡xtan⁡xf'(x) = 3 \sec x \tan x
Evaluate at x = 0: Since sec 0 = 1 and tan 0 = 0, the slope at x = 0 is zero.
f′(0)=3(1)(0)=0f'(0) = 3(1)(0) = 0
Answer: f′(x)=3sec⁡xtan⁡xf'(x) = 3\sec x \tan x, and the slope at x=0x = 0 is 00.

Why It Matters

This derivative appears frequently in AP Calculus when you integrate sec⁡xtan⁡x\sec x \tan x or differentiate expressions involving secant. It also shows up in related-rates and optimization problems that involve trigonometric models, such as the angle of a spotlight beam or the slope of a ramp.

Common Mistakes

Mistake: Writing the derivative as sec x sec x (i.e., sec²x) by confusing it with the derivative of tan x.
Correction: The derivative of tan x is sec²x. The derivative of sec x is sec x tan x. Keep these two results distinct by noting that each trig derivative pairs with a different cofactor.

Related Terms