3.4 Inverse Trig Derivatives

The formulas

d/dx[arcsin x] = 1/√(1−x²), d/dx[arccos x] = −1/√(1−x²), d/dx[arctan x] = 1/(1+x²).

Notice

arccos differs from arcsin only by a negative sign. arctan has 1 + x² (no square root) in the denominator.

Key terms

d/dx arcsin x

= 1/√(1−x²).

d/dx arccos x

= −1/√(1−x²).

d/dx arctan x

= 1/(1+x²).

arcsin vs arccos

Same magnitude, opposite sign.

arctan denominator

1 + x² (no radical).

Try an AP-style question

d/dx arcsin x = ?

1. 1/(1 + x²)
2. 1/√(1 − x²)
3. −1/√(1 − x²)
4. 1/(x√(x² − 1))