Derivatives of Inverse Trigonometric Functions: Exam Tips

Just as sin x and cos x have derivatives, so do their inverses: arcsin x, arccos x, arctan x, and the rest. These derivatives of inverse trigonometric functions are algebraic — no trigonometry remains in the answer. For example, d/dx arcsin x = 1/√(1 − x²).

They matter for two reasons. First, they appear directly whenever you differentiate expressions like arctan(2x). Second, and more famously, they run in reverse: integrals that produce inverse trig functions are recognized by spotting these exact derivative patterns. They are also essential in problems about angles changing with time.

Key Concepts

• Inverse trig functions: arcsin x (also written sin⁻¹x) is the angle whose sine is x, defined for x between −1 and 1.
• Algebraic results: each derivative is a rational expression or square root — the trigonometry differentiates away.
• Domains: arcsin and arccos derivatives need |x| < 1, because √(1 − x²) is zero at the endpoints.
• Sign pairs: arccos, arccot, and arccsc carry minus signs; arcsin, arctan, and arcsec are positive.

Formulas

• d/dx arcsin x = 1/√(1 − x²), for |x| < 1.
• d/dx arccos x = −1/√(1 − x²), for |x| < 1.
• d/dx arctan x = 1/(1 + x²), for all real x.
• d/dx arccot x = −1/(1 + x²).
• With chain rule: d/dx arcsin(u) = u'/√(1 − u²); d/dx arctan(u) = u'/(1 + u²).

Worked Example

Differentiate y = arctan(2x) and evaluate y'(1/2).

The outer function is arctan(u) with u = 2x. Its derivative with respect to u is 1/(1 + u²), so by the chain rule y' = 1/(1 + (2x)²) · d/dx(2x) = 1/(1 + 4x²) · 2 = 2/(1 + 4x²). Now evaluate at x = 1/2: the denominator is 1 + 4·(1/2)² = 1 + 4·(1/4) = 1 + 1 = 2. So y'(1/2) = 2/2 = 1. A quick sanity check with secants: arctan(1) = π/4 ≈ 0.78540, and arctan(1.02) ≈ 0.79540; the difference 0.01 over an x-step of 0.01 gives a slope of about 1.0, matching the exact derivative.

Common Misconceptions

• The arcsin derivative works at x = ±1. It does not — the denominator √(1 − x²) becomes zero there, so the derivative is undefined at the endpoints.
• arccos has the same derivative as arcsin. It has the same magnitude but a minus sign: d/dx arccos x = −1/√(1 − x²).
• sin⁻¹x means 1/sin x. It does not. sin⁻¹x is the inverse function (arcsin), while 1/sin x is the reciprocal (cosecant). The notation is a classic trap.

Real-World Applications

• Integration: recognizing 1/(1 + x²) as the arctan derivative is how whole families of integrals are solved.
• Tracking problems: the rate at which a camera angle changes to follow a moving object involves arctan derivatives.
• Navigation: bearing and heading calculations differentiate inverse trig expressions.
• Optics: Snell's law rearranged with arcsin differentiates to give angle sensitivity.

In short, inverse trig derivatives trade trigonometry for algebra: arcsin gives 1/√(1 − x²), arctan gives 1/(1 + x²), and the co-functions take minus signs. Watch the domains, apply the chain rule for inner functions, and never confuse sin⁻¹x with 1/sin x.