Inverse Trig Derivatives
Derivative formulas for arcsin, arccos, arctan and other inverse trigonometric functions.
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