CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03

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Principal arcsine has no finite derivative at $-1$ or $1$

Example

The principal arcsine $\arcsin:[-1,1]\to[-\pi/2,\pi/2]$ has no finite derivative at either endpoint $1$ or $-1$ (with the library's relative, one-sided endpoint convention).

Facts & Assumptions

Given: No hypotheses beyond those quantified in the statement.

[L1]

On $[-1,1]$ , $\sin(\arcsin y)=y$ , and $\arcsin(1)=\pi/2$ , $\arcsin(-1)=-\pi/2$ (Principal inverse sine and inverse cosine).

[L2]

The chain rule applies to derivatives relative to their domains (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$ ).

[L3]

$(\sin x)'=\cos x$ , while $\cos(\pi/2)=\cos(-\pi/2)=0$ (The derivatives of sine and cosine are cosine and minus sine, Quarter-turn values and shifts by pi/2 and pi, Pythagorean and parity identities for all six trigonometric functions on their natural domains).

Proof

technique · contradiction

1.1

Suppose $\arcsin$ had a finite derivative at $1$ . Differentiate the identity $\sin(\arcsin y)=y$ at $1$ relative to $[-1,1]$ . The derivative of the right side is $1$ , whereas [L2] and [L3] make the derivative of the left side $\cos(\pi/2)(\arcsin)'(1)=0$ , a contradiction.

assume-contraL1L2L3

1.2

The identical argument at $-1$ gives $1=\cos(-\pi/2)(\arcsin)'(-1)=0$ .

assume-contraL1L2L3

2.1

Therefore neither finite endpoint derivative exists.

step 1.1step 1.2discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 70 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, §4.24(ii) Derivatives (standard reference, not scraped)

Principal arcsine has no finite derivative at −1-1−1 or 111