Inverse Trig Derivatives
Reference for derivatives of inverse trig functions: arcsin, arccos, arctan, arccot, arcsec, and arccsc.
Includes domain restrictions and worked examples.
The Formulas
d/dx [arcsin(x)] = 1 / √(1 - x²)
d/dx [arccos(x)] = -1 / √(1 - x²)
d/dx [arctan(x)] = 1 / (1 + x²)
d/dx [arccot(x)] = -1 / (1 + x²)
d/dx [arcsec(x)] = 1 / (|x|√(x² - 1))
d/dx [arccsc(x)] = -1 / (|x|√(x² - 1))
d/dx [arccos(x)] = -1 / √(1 - x²)
d/dx [arctan(x)] = 1 / (1 + x²)
d/dx [arccot(x)] = -1 / (1 + x²)
d/dx [arcsec(x)] = 1 / (|x|√(x² - 1))
d/dx [arccsc(x)] = -1 / (|x|√(x² - 1))
These formulas give the rate of change of each inverse trig function. They appear frequently in calculus, especially in integration problems.
Variables
| Symbol | Meaning |
|---|---|
| x | The input variable |
| d/dx | Derivative with respect to x |
| arcsin, arccos, arctan | Inverse trig functions (also written sin⁻¹, cos⁻¹, tan⁻¹) |
Example 1
Find d/dx [arctan(3x)]
Using chain rule: d/dx [arctan(u)] = (1/(1+u²)) × du/dx
u = 3x, du/dx = 3
= 3 / (1 + 9x²)
Example 2
Find d/dx [arcsin(x/2)]
u = x/2, du/dx = 1/2
= (1/√(1 - (x/2)²)) × (1/2)
= 1 / (2√(1 - x²/4)) = 1 / √(4 - x²)
When to Use Them
Use inverse trig derivatives when:
- Differentiating expressions containing arcsin, arccos, or arctan
- Recognizing integral forms that result in inverse trig functions
- Solving related rates or optimization problems in calculus
- Working with angles defined implicitly in physics or engineering
Key Notes
- arcsin and arccos have domain [−1, 1] — their derivatives blow up to ±∞ as x → ±1 because the inverse sine curve becomes vertical at those endpoints; arctan and arccot are defined for all real x
- The derivatives of arcsin and arccos are negatives of each other: d/dx[arccos(x)] = −d/dx[arcsin(x)] — this is because arcsin(x) + arccos(x) = π/2 always, so their derivatives must sum to zero
- These formulas are the key to recognizing integration patterns: ∫ 1/√(1−x²) dx = arcsin(x) + C and ∫ 1/(1+x²) dx = arctan(x) + C — spotting these forms on sight is a core integration skill
- With the chain rule applied: d/dx[arctan(u)] = (du/dx)/(1+u²) — always substitute u = inner function, compute du/dx, then divide by (1 + u²); failing to apply the chain rule is the most frequent error