Calculus — Derivatives⏱ 5 min read

Inverse Trig Derivatives

d/dx(arcsin x) = 1/√(1−x²)

What is the Inverse Trig Derivatives?

These formulas give the derivatives of the inverse trigonometric functions, and arise naturally whenever an inverse trig function needs differentiating — they also appear frequently as the results of certain integrals.

arcsin and arccos have the same derivative magnitude but opposite signs. That's not a coincidence: since arcsin(x) + arccos(x) = π/2 (a constant) for all valid x, their derivatives must sum to zero.

What Each Variable Means

arcsin x

Inverse sineDefined only for |x| < 1; derivative is 1/√(1−x²).

arccos x

Inverse cosineDefined only for |x| < 1; derivative is −1/√(1−x²).

arctan x

Inverse tangentDefined for all real x; derivative is 1/(1+x²).

When to Use It

Differentiating any expression involving arcsin, arccos, or arctan
Combined with the chain rule when the inverse trig function's argument is itself a function of x
Recognizing certain integral results that produce inverse trig functions

Step-by-Step Examples

Example 1: arcsin with the chain rule

Problem: Differentiate y = arcsin(3x)

Identify outer and inner functions

Outer: arcsin(u). Inner: u = 3x.

d/du[arcsin u] = 1/√(1-u²), du/dx = 3

Multiply by du/dx = 3

Apply the chain rule, substituting u = 3x back in.

dy/dx = 3 / √(1 − (3x)²)

✓

Answer: dy/dx = 3 / √(1 − 9x²)

Example 2: arctan with the chain rule

Problem: Differentiate y = arctan(x²)

Identify outer and inner functions

Outer: arctan(u). Inner: u = x².

d/du[arctan u] = 1/(1+u²), du/dx = 2x

Multiply the two derivatives

Substitute u = x² back in.

dy/dx = 2x · 1/(1 + (x²)²)

✓

Answer: dy/dx = 2x / (1 + x⁴)

Interactive Calculator

x value (between -1 and 1)

Result will appear here

Common Mistakes

Mistake: Forgetting the domain restriction |x| < 1 for arcsin and arccos.
Fix: arcsin x and arccos x — and their derivatives — are only defined for |x| < 1. arctan x, by contrast, is defined for all real numbers.
Mistake: Forgetting to apply the chain rule when the argument isn't just x.
Fix: d/dx[arcsin(3x)] is not simply 1/√(1-x²) — you must also multiply by the derivative of the inner function (3x)' = 3.