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Inverse Trig Derivatives

d/dx(arcsin x) = 1/√(1−x²)

📖 Inverse Trig Derivative Formulas

These formulas arise naturally when differentiating inverse trigonometric functions. They appear frequently in integration as well.

Function Derivative Domain
arcsin x 1 / √(1 − x²) |x| < 1
arccos x −1 / √(1 − x²) |x| < 1
arctan x 1 / (1 + x²) All real x
arccot x −1 / (1 + x²) All real x

📝 Example — Using Chain Rule with arcsin

Differentiate: y = arcsin(3x)

1
Identify outer and inner functionsOuter: arcsin(u)    Inner: u = 3x
2
Derivative of arcsin(u) with respect to u1/√(1 − u²)
3
Multiply by du/dx = 3 (chain rule)dy/dx = 3 / √(1 − (3x)²)
✓
Answer: dy/dx = 3 / √(1 − 9x²)

📝 Example — arctan

Differentiate: y = arctan(x²)

1
Outer: arctan(u), inner: u = x²d/du(arctan u) = 1/(1 + u²)
2
Multiply by du/dx = 2x (chain rule)dy/dx = 2x · 1/(1 + (x²)²)
✓
Answer: dy/dx = 2x / (1 + x⁴)

🧮 Calculator

Compute derivative values of arcsin and arctan at a given x.

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🔗 Related Formulas

🎯 Quick Fact

Notice that arcsin and arccos have the same derivative, but opposite signs. They must — because arcsin(x) + arccos(x) = π/2 (a constant), so their derivatives must sum to zero.