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Calculus learning

Derivative of arcsin(x): Inverse Function Proof and Endpoint Limits

The derivative of arcsin(x) is a reciprocal square root. The distinction between the function’s domain and its derivative’s domain matters: arcsin exists at ±1, but its derivative grows without bound as an endpoint is approached from the interior.

Direct answer

d/dx arcsin(x)=1/√(1−x²), for −1<x<1.

The real function is defined on [−1,1], but a finite two-sided derivative is only available in the interior.

Derivative of arcsin(x) worked method

Let y=arcsin x. Then sin y=x and the principal angle y lies between −π/2 and π/2.

Implicit differentiation gives cos y·y′=1. In the interior cos y is positive.

Using cos y=√(1−sin²y)=√(1−x²) gives the derivative.

For arcsin(x/2), differentiate the inside first: (1/2)/√(1−x²/4). Its derivative domain is −2<x<2.

1
sin y=x → cos y · y′=1
2
y′=1/√(1−x²)
3
(arcsin(g(x)))′=g′(x)/√(1−g(x)²)

Worked examples

d/dx arcsin(x)
11−x2\frac{1}{\sqrt{1 - x^{2}}}

Interior domain only.

d/dx arcsin(x/2)
12 1−x24\frac{1}{2\,\sqrt{1 - \frac{x^{2}}{4}}}

Require |x|<2.

d/dx arcsin(x²)
2 x1−x4\frac{2\,x}{\sqrt{1 - x^{4}}}

The formula is finite on |x|<1.

Common mistakes

  • • Confusing arcsin with 1/sin.
  • • Using the formula at endpoints.
  • • Omitting the inner derivative.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

The engine accepts asin(x). Keep the stricter derivative domain even when the function itself is defined at endpoints.

求导:asin(x)
Open the calculator without a preset: 求导

References

Frequently asked questions

What does sin⁻¹ mean here?+

In inverse-function notation it means arcsin, not the reciprocal csc.

Why the positive square root?+

The principal arcsin range makes cosine nonnegative, and positive in the interior.

What spelling is accepted by the preset?+

asin(x) is the parser’s inverse-sine spelling.

Conclusion

Derivative of arcsin(x) starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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