---
tag: 0GA6G43EA9
kind: statement
role: proposition
name: Derivative of arcsine
msc: [33B10, 26A24]
uses: [0B89D93YA1, 0M0RZ3HSB0, 00R4MWW532, 03H7WZ4T3K]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions
    locator: Differentiating the inverse sine function
---

## Statement

The arcsine is differentiable on $(-1, 1)$, and
$$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \quad \text{for } -1 < x < 1.$$

## Remarks

Let $y = \arcsin x$. The [[inverse function rule|0M0RZ3HSB0]] gives the derivative $\frac{1}{\cos y}$. By the [[Pythagorean identity|03H7WZ4T3K]], $\cos y = \sqrt{1 - x^2}$, since $\cos y > 0$ for $|y| < \pi/2$.
The arcsine is not differentiable at $\pm 1$. Its graph has vertical tangents there.
Similarly, $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}$. Mathlib states the main claim as `Real.hasDerivAt_arcsin`.

## Explanations

### L1 human

The derivative of the arcsine is an algebraic function, although the arcsine itself is not.
It becomes infinite at the ends of the interval $[-1, 1]$.