differentiable

Def 31.3

$F$ is differentiable at $x$ if

$$(\exists a)(\forall \epsilon >0)(\exists \delta )|x-y|<\delta ⟹|\frac{F(y)-F(x)}{y-x}-a|<\epsilon$$

where we can say $a=F^{\prime }(x)$

Thm 31.4

differentiable $f$ on $(a,b)$ and $f^{\prime }(x)$ bounded $⟹$ $f$ is Lipschitz

Thm 32.1 Fundamental Theorem

1. ${\int }_a^x{g}^{\prime }(t)dt=g(x)-c$
2. if $F^{\prime }$ is integrable, $F(b)-F(a)={\int }_a^b{F}^{\prime }(t)dt$

Thm 32.2 Mean Value Theorem(MVT)

let $f$ be continue on $[a,b]$ and $f$ is differentiable on $(a,b)$

$$(\exists \xi \in (a,b))\frac{f(b)-f(a)}{b-a}=f^{\prime }(\xi )$$

Rolle’s Lemma

Let $f$ be continue on $[a,b]$ and differentiable on $(a,b)$

$$f(a)=f(b)⟹(\exists \xi )f^{\prime }(\xi )=0$$

Rules

1. $(f+\lambda g{)}^{\prime }=f^{\prime }+\lambda g^{\prime }$
2. $(f\circ g{)}^{\prime }=(f^{\prime }\circ g)g^{\prime }$
3. $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$
4. $f\in [a,b]$ is strict monotone, $f^{-1}$ is continue

Thm 40.1 Chain rule

$$(f\circ g{)}^{\prime }=f^{\prime }(g(x))g^{\prime }(x)$$

Prop 40.1

Let $g:\mathbb{R}⟶\mathbb{R}$

$$\frac{d}{dy}{\int }_a^{b}g(y,t)dt={\int }_a^b\frac{d}{dy}g(y,t)dt$$

Thm 40.2 differential

$$H^{\prime }(a)=\underset{n}{\lim }\frac{H(a+{\xi }_n)-H(a)}{{\xi }_n}$$

Thm

Inverse of differentiable function is differentiable.

Thm

$$\frac{f(x)-f(x_0)}{g(x)-g(x_0)}=\frac{f^{\prime }(\xi )}{g^{\prime }(\xi )}$$

$x_0<\xi <x$

Thm

$h^{\prime }$ is continue $⟺$ $h$ is continue and differentiable.