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Derivatives of Inverse Trig Calculator

Online inverse trig derivative calculator for arcsin, arccos, arctan and arccot. Enter x to instantly get the derivative formula and exact value, with derivation steps and domain notes—ideal for calculus practice and homework.

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(arcsin x)′=1/√(1−x²) (arctan x)′=1/(1+x²)
Inverse trig
|x|≤1 for arcsin/arccos

📖 Tutorial｜Derivatives of Inverse Trig

1. Definition
Inverse trig derivatives: (arcsin x)′=1/√(1−x²), (arccos x)′=−1/√(1−x²), (arctan x)′=1/(1+x²), (arccot x)′=−1/(1+x²).
2. Symbols
FunctionDerivative
arcsin x1/√(1−x²)
arccos x−1/√(1−x²)
arctan x1/(1+x²)
arccot x−1/(1+x²)
3. How it works
  • We apply the analytic formula;
  • arcsin/arccos require |x|≤1;
  • Then substitute x.
4. Steps
  1. Choose an inverse trig function;
  2. Enter x (mind the domain);
  3. Click Calculate.
5. Example
Example: arcsin x at x=0.5. Derivative=1/√0.75≈1.1547.
6. Pitfalls
arcsin and arccos require |x|≤1;
arccos differs from arcsin by a minus sign;
arctan accepts all real x.

❓ FAQ｜Derivatives of Inverse Trig

How is arcsin′ derived?
Let y=arcsin x so sin y=x. Differentiate: cos y·y′=1, so y′=1/cos y=1/√(1−x²).
How are arccos and arcsin related?
They differ only by a minus sign: arccos x = π/2 − arcsin x.
Why is arctan′ so simple?
From tan y=x, sec²y·y′=1 and sec²y=1+x², so y′=1/(1+x²).
Why require |x|≤1?
arcsin and arccos are defined only on [−1,1].
Which inverse trig derivatives are most tested?
arcsin′ and arctan′ appear most often—memorize them.