Chapter 4.
1. Three equivalent definitions of differentiable and derivative: Let f:D→Rwith x0an accumulation
of D, and x0∈D. The following are three are equivalent definitions of differentiable:
(a) The limit
L:= lim
x→x0
f(x)−f(x0)
x−x0
exists.
(b) The limit
L:= lim
t→0
f(x0+t)−f(x0)
t
exists.
(c) There exists a number Land a function φsuch that limt→0φ(t) = 0 and
f(x0+t) = f(x0) + Lt +tφ(t),∀t∈ {t|t+x0∈D}
The limit L, if it exists, is called the derivative of fat x0and is denoted by f0(x0). If a function has a
derivative at each x∈Sthen we say the function is differentiable on S. If a function is differentiable
on its domain, we simply say that the function is differentiable.
2. Let f:D→R,x0∈Dsuch that there exist a neighborhood Qof x0where ∀x∈Q∩Dwe have
f(x)≥( resp. ≤)f(x0), then we say that fhas a relative minimum (resp. relative maximum) at x0.
Note, a relative maximum or a relative minimum can be referred to as a relative extremum.
3. Chain Rule.
4. Rolle’s Theorem.
5. Mean Value Theorem (for derivatives).
6. Cauchy Mean Value Theorem.
7. L’Hˆopital’s Rule.
8. Inverse Function Theorem: Let Dbe an interval and f:D→Rbe a differentiable function such
that ∀x∈Dwe have f0(x)6= 0. Then fis a one-to-one function from Donto an interval J. Its
inverse function f−1:J→Dexists and is differentiable and for all y∈Jwe have
¡f−1¢
0
(y) = 1
f0(x),where y=f(x).
