Remaining trigonometric derivatives
Differentiate cotangent, secant, and cosecant.
Remaining trigonometric derivatives is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why remaining trigonometric derivatives works
Each of these comes out of the quotient rule applied to a ratio of sine and cosine. Differentiating one over cosine produces sine over cosine squared, which regroups as secant times tangent, and the same computation on the two co-functions produces their results, with the negative signs coming from differentiating a sine in the denominator.
What each symbol means
$x$ is measured in radians.
Remaining trigonometric derivatives: when it holds
Each function and the expression on the right must be defined at the point.
When it stops applying
Each statement is silent wherever its own function has an asymptote. Cotangent and cosecant have no value at any whole multiple of π, and secant has none at the odd multiples of π/2, so no derivative exists at those inputs.
Remaining trigonometric derivatives: a worked example
$\frac{d}{dx}\sec(2x)=2\sec(2x)\tan(2x)$.
The mistake to avoid
What people do: Giving secant the same derivative as tangent.
Why it goes wrong: Secant squared belongs to tangent alone; secant itself differentiates to secant times tangent, which behaves very differently and even changes sign in different quadrants.
Do this instead: Rebuild the answer from one over cosine with the quotient rule whenever the pair gets mixed up.
Remaining trigonometric derivatives: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x$ is measured in radians.
- Check the conditions before substituting. Each function and the expression on the right must be defined at the point.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where remaining trigonometric derivatives comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about remaining trigonometric derivatives
Why do cotangent and cosecant carry negative signs?
Both are built with sine in the denominator, and differentiating that denominator drags a negative sign into every term of the result.
Is there a way to keep these three straight?
Pair them up. Tangent goes with secant squared and secant goes with secant times tangent, then each co-function is the same expression with its own co-partners and a negative sign.
What is the derivative of the secant of 2x?
Multiply by the inner derivative, giving 2 times the secant of 2x times the tangent of 2x.
Do these three also assume radians?
Yes. All six trigonometric derivatives rest on the same radian limit for sine, so working in degrees rescales every one of them.