Derivative of sec²(x)
The derivative of $\sec^2 x = \dfrac{1}{\cos^2 x}$ is $\dfrac{d}{dx}\sec^2 x = 2\sec^2 x\tan x = \dfrac{2\sin x}{\cos^3 x}$. Because $\sec^2 x$ is itself the derivative of $\tan x$, this is also the second derivative of $\tan x$.
See the derivative of sec²(x) step by step in the calculator →
📈 Graph of sec²(x) and its derivative
🔍 Why is the derivative of sec²(x) equal to $2\sec^2 x\tan x$?
$\sec^2 x = (\sec x)^2$ – outer function $u^2$, inner function $\sec x$ with $(\sec x)' = \sec x\tan x$:
Alternatively with $(\cos x)^{-2}$: $\;-2(\cos x)^{-3}\cdot(-\sin x) = \frac{2\sin x}{\cos^3 x}$ – the same function.
🔗 Derivative of sec²(x) with an inner function
If the argument is a function $u(x)$, the chain rule adds the inner derivative:
✏️ Worked examples
| Function | Derivative | Note | |
|---|---|---|---|
| $\sec^2(2x)$ | $4\sec^2(2x)\tan(2x)$ | Inner derivative $2$ times the factor $2$ from the square. | steps → |
| $\tan^2 x$ | $2\sec^2 x\tan x$ | The same derivative as $\sec^2 x$, because $\sec^2 x = 1 + \tan^2 x$. | steps → |
| $\frac{1}{\cos^2 x}$ | $\frac{2\sin x}{\cos^3 x}$ | Same function, written with cosine. | steps → |
| $x\sec^2 x$ | $\sec^2 x + 2x\sec^2 x\tan x$ | Product rule. | steps → |
⚠️ Common mistakes
Writing $2\sec x\tan x$ – one factor $\sec x$ is missing.
Thinking $\sec^2 x$ means $\sec(x^2)$.
❓ Frequently asked questions
What is the second derivative of tan(x)?
$(\tan x)'' = (\sec^2 x)' = 2\sec^2 x\tan x$.
Why do sec²(x) and tan²(x) have the same derivative?
Because $\sec^2 x - \tan^2 x = 1$ is constant.
What is the derivative of 1/cos²(x)?
$\frac{2\sin x}{\cos^3 x} = 2\sec^2 x\tan x$.