notes / Math 55b / chapter 5: Analysis in One Variable / From my notebook
N22 Differentiation
Lecture 19
Definition N22.1. is differentiable at if
exists, i.e. for all there is such that
Proposition N22.2. If is differentiable at , then is continuous at . (The converse is false.)
Proof.
and multiplication is continuous, so . ∎
The usual rules of calculation (, , etc.) hold.
Theorem N22.3 (Mean value theorem). If is differentiable, then there is such that
Corollary N22.4 (Mean value inequality). If for all , then
Generalization—Taylor’s theorem:
Theorem N22.5 (Taylor’s theorem). Let be times differentiable. The degree Taylor polynomial of at is
Then there is such that