General bound: If |f'(x)| <= M for all x in I, then |f(x1) - f(x2)| <= M|x1 - x2| (Lipschitz condition).
Common bounds derived via LMVT:
| Inequality | Function | Derivative bound |
|---|---|---|
| |sin a - sin b| <= |a-b| | sin x | |cos x| <= 1 |
| |tan a - tan b| >= |a-b| (a,b in (-pi/2,pi/2)) | tan x | (x) >= 1 |
| | - | >= |a-b| for a,b >= 0 | >= 1 for x >= 0 | |
| |ln a - ln b| <= |a-b|/min(a,b) | ln x | 1/x, with x >= min(a,b) |
| |sqrt(a) - sqrt(b)| <= |a-b|/(2*sqrt(min(a,b))) | sqrt(x) | ) |