Remaining inverse-trig derivatives: what it means and how to use it
Differentiate inverse cosine and the common decreasing convention for inverse cotangent.
Remaining inverse-trig derivatives is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why remaining inverse-trig derivatives works
Arcsine and arccosine always add to π/2, a constant, so their rates of change must cancel exactly, which is the entire reason arccosine picks up a negative sign. The same fixed-sum relationship between arctangent and inverse cotangent explains the second formula.
What each symbol means
$x$ is the real input.
Remaining inverse-trig derivatives: when it holds
For arccos, $|x|<1$; inverse-cotangent conventions can vary by textbook.
When it stops applying
The arccosine rule needs the input strictly between −1 and 1, since at the two ends the graph turns vertical and the denominator hits 0. The inverse cotangent rule depends on convention too: some books define it as the arctangent of the reciprocal, which is undefined at 0 and jumps there.
Remaining inverse-trig derivatives: a worked example
$\frac{d}{dx}\arccos(3x)=-3/\sqrt{1-9x^2}$.
The mistake to avoid
What people do: Dropping the negative sign because the arcsine version was learned first.
Why it goes wrong: Arccosine falls from π down to 0 across its domain, so a positive derivative would contradict its own graph.
Do this instead: Glance at the shape: a function that decreases everywhere must have a negative derivative everywhere it is defined.
Remaining inverse-trig 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 the real input.
- Check the conditions before substituting. For arccos, $|x|<1$; inverse-cotangent conventions can vary by textbook.
- 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 inverse-trig derivatives comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules lesson — 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 inverse-trig derivatives
Why is the arccosine derivative just the arcsine one negated?
Because the two functions always add to the constant π/2. Differentiating a constant gives 0, so the two rates must be exact opposites.
Does arccosine have a derivative at the input 1?
No. The square root in the denominator becomes 0 there, which matches the vertical tangent at that end of the graph.
Is there a derivative for the inverse secant as well?
Yes: one over the absolute value of x times the root of x squared minus 1, defined only when the input is beyond 1 or below −1.
How do I differentiate the arccosine of 3x?
Apply the chain rule and keep the negative sign, giving −3 over the root of 1 minus 9x², valid while the input stays between −1/3 and 1/3.