Reciprocal trigonometric functions

Derivative of sec(x): Formula, Proof, and Examples

Secant is the reciprocal of cosine, not an inverse trigonometric function. Rewriting it as 1/cos(x) makes both the sign and the product sec(x)tan(x) transparent.

Direct answer

ddx[sec⁡x]=sec⁡x tan⁡x\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec x\right] = \sec x\,\tan x

Requires cos(x) ≠ 0; angles are measured in radians.

When does a secant derivative need the chain rule?

Read the expression from the outside inward. The secant rule supplies sec(u)tan(u); the chain rule is needed only when its input u also changes with x.

Ask first 1

Is the expression exactly sec(x)?

What to do

Use sec(x)tan(x), then keep the restriction cos(x)≠0.

Ask first 2

Is sec applied to an inner expression u=g(x)?

What to do

Keep sec(g(x))tan(g(x)) and multiply by g′(x).

Ask first 3

Is sec(g(x)) multiplied by another function?

What to do

Use the product rule at the top level and the chain rule inside the secant factor.

Proof from sec(x) = 1/cos(x)

Write sec(x)=[cos(x)]⁻¹ and differentiate with the power and chain rules.

The two negative signs cancel: −[cos(x)]⁻² · [−sin(x)] = sin(x)/cos²(x).

Split the quotient as [1/cos(x)]·[sin(x)/cos(x)] to obtain sec(x)tan(x).

1
[cos(x)⁻¹]' = sin(x)/cos²(x)
2
= [1/cos(x)] [sin(x)/cos(x)]
3
= sec(x)tan(x)

Worked examples

ddx[sec⁡(2 x)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec\left(2\,x\right)\right]
2 sec⁡(2 x) tan⁡(2 x)2\,\sec\left(2\,x\right)\,\tan\left(2\,x\right)

Multiply by the inner derivative 2.

ddx[sec⁡(x2)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec\left(x^{2}\right)\right]
2x sec(x²)tan(x²)

Apply the chain rule to x².

ddx[sec⁡3x]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec^{3} x\right]
3 sec3 x tan⁡x3\,\mathrm{sec}^{3}\,x\,\tan x

Combine the power rule with sec′(x).

Common mistakes

  • • Confusing sec(x)=1/cos(x) with arccos(x), the inverse of cosine.
  • • Dropping one of the two negative signs in the reciprocal-cosine derivation.
  • • Ignoring points where cos(x)=0.

Error clinic: find the missing factor

Incorrect attempt

[sec(2x)]′ = sec(2x)tan(2x)

Why it fails

The secant rule is present, but the inner derivative of 2x is missing.

Correction

[sec(2x)]′ = 2sec(2x)tan(2x)

Incorrect attempt

[sec(x²)]′ = 2x sec(x)tan(x)

Why it fails

The inner expression was changed after differentiating. Both outer factors must keep x² as their input.

Correction

[sec(x²)]′ = 2x sec(x²)tan(x²)

Practice the same idea with a new inner function

Write the preserved secant input first, then multiply by the derivative of that input. Reveal the answer only after you have identified both pieces.

ddx[sec⁡(3 x−1)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec\left(3\,x - 1\right)\right]

Check before solving

Identify the constant inner derivative.

Reveal the reference answer

Reference answer

3 sec⁡(3 x−1) tan⁡(3 x−1)3\,\sec\left(3\,x - 1\right)\,\tan\left(3\,x - 1\right)
ddx[sec⁡(x2+1)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec\left(x^{2} + 1\right)\right]

Check before solving

Preserve x²+1 in both outer factors.

Reveal the reference answer

Reference answer

2x sec(x²+1)tan(x²+1)

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Enter sec(2x) directly. The calculator recognizes it as 1/cos(2x), keeps the cosine restriction visible, and applies the chain rule.

Derivatives:sec⁡(2 x)\sec\left(2\,x\right)
Open the calculator without a preset: Derivatives

Frequently asked questions

Is sec⁻¹(x) the same as 1/sec(x)?+

The notation is ambiguous. In inverse-function notation sec⁻¹ can mean arcsec, while the reciprocal of sec(x) is cos(x). Write the intended function explicitly.

Why is the derivative undefined when cos(x)=0?+

Sec(x)=1/cos(x) is already undefined there, so its derivative cannot be defined at those points either.

Can I enter sec(x) directly in the calculator?+

Yes. The input is recognized as the equivalent reciprocal-cosine expression before the deterministic derivative rules are applied.

Conclusion

Differentiate secant from the outside inward: keep the complete input inside sec and tan, multiply by the inner derivative, and retain the original cosine restriction.

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