Hello to everyone

I have a question with many parts that i am needing to do.

(a)Derive the 2point formula, for numerical differentiation:

$\displaystyle f'(x) = \frac{f(x + h) - f(x)}{h} + O(h) $

What is $\displaystyle O(h) \ $?

(b) Derive the same formula as (b) witht 3points formula

$\displaystyle f'(x) = \frac{f(x + h) - f(x-h)}{2h} + O(h^2). $

(c) Discretisation error in (a) and (b) is

$\displaystyle E_2(h) = |f'(x) - \frac{f(x + h) - f(x)}{h} |$

$\displaystyle E_3(h) = |f'(x) - \frac{f(x + h) - f(x-h)}{h^2} |$

Prove that discretisation error for f'(1), where f(x) = exp(x) is

$\displaystyle E_2(h) = e|1 - \frac{e^h - 1}{h}| = e\frac{h}{2} + O(h^2)$

$\displaystyle E_3(h) = e|1 - \frac{e^h - e^-h}{2h}| = e\frac{h^2}{6} + O(h^3)$

I am knowing that

$\displaystyle f'(x) = \lim_{h \to 0} ~ \frac{f(x + h) - f(x)}{h} $

but I cannot realise how to do these parts? Can someone please showing me how to prove?