Derivative of sec(x)
The derivative of the secant is $\dfrac{d}{dx}\sec x = \sec x\tan x = \dfrac{\sin x}{\cos^2 x}$, defined wherever $\cos x \neq 0$.
$$ \frac{d}{dx}\left(\sec x\right) = \sec x\tan x $$
See the derivative of sec(x) step by step in the calculator →
📈 Graph of sec(x) and its derivative
🔍 Why is the derivative of sec(x) equal to $\sec x\tan x$?
Write $\sec x = (\cos x)^{-1}$ and use the chain rule:
$$ (\sec x)' = -1\cdot(\cos x)^{-2}\cdot(-\sin x) = \frac{\sin x}{\cos^2 x} = \frac{1}{\cos x}\cdot\frac{\sin x}{\cos x} = \sec x\tan x. $$
🔗 Derivative of sec(x) with an inner function
If the argument is a function $u(x)$, the chain rule adds the inner derivative:
$$ \frac{d}{dx}\sec\big(u(x)\big) = u'(x)\sec\big(u(x)\big)\tan\big(u(x)\big) $$
✏️ Worked examples
⚠️ Common mistakes
Writing $\tan x$ alone or $\sec^2 x$ – that is the derivative of $\tan x$.
Sign errors when differentiating $(\cos x)^{-1}$: the two minus signs cancel.
❓ Frequently asked questions
What is the derivative of 1/cos(x)?
$\frac{\sin x}{\cos^2 x}$, which is $\sec x\tan x$.
What is the derivative of sec²(x)?
$2\sec^2 x\tan x$ – see the separate page.
Where is the derivative of sec(x) zero?
Where $\tan x = 0$, i.e. at $x = k\pi$.