Intaai wowonod montok do Pongungaban

'''Pongungaban''' (Boros Inggilis: ''differentiation'') nopo nga' iso katandaan id raan gana' pongintaban do [[Sidsimban]] (''calculus''). Katandaan diti sambalik do [[Ponompuuvan]] (''integration'').

== Poomitanan ==
Montok iso pampos (''function'') <math>f(x) = x^3 + 2x</math>, tungab (''derivative'') dau nopo nga' suaton do <math>f'(x)</math> toi <math>\frac{d}{dx}(f(x))</math>. It koiso nopo nga' roitan do wayagait di Newton, om it koduo wayagait di Leibniz.

Momoguno do tuku sundu (''power rule''), ii tungab nopo nga:

<math>f'(x) = 3x^2 + 2</math>

== Pongomoi ==
Mamakai do pongomoi dompok <math>\varepsilon</math>-<math>\delta</math>, montok iso pampos kigatang totunud (''real-valued function''), karait tokou id sogom <math>a</math> suang raung dau, om haro id suang raung dau tokuri sonsopo poingukab (''open interval'') poingion di sogom <math>a</math>, om hino no i dompok <math>L</math> poingkaa

<math>L = \lim_{\Delta x \rightarrow 0} \frac{f(a+\Delta x) -f(a)}{\Delta x}</math>

Montok tikid tintob totunud <math>\varepsilon</math> haro nogi iso tintob totunud danan (''positive real number''), <math>\delta</math> moi do montok tikid <math>\Delta x</math> om poingkaa:

<math>|\Delta x| < \delta \quad \textrm{om} \quad \Delta x \neq 0</math>

Sumusuut oworun no daa <math>f(a+\Delta x) </math> om <math>\left| L - \frac{f(a + \Delta x) - f(a)}{\Delta x} \right| < \varepsilon</math>.

== Tuku pongungaban ==

Haro piipiro tuku<ref>https://openstax.org/books/calculus-volume-1/pages/3-3-differentiation-rules</ref> do popouhan pongihiman do tungab di pampos timpou otungab. Id saau, <math>x</math> nopo nga' iso '''osimban''' (''variable''), om suai pimato pakayon nopo nga' '''togom''' (''constant''). Nga' montok ii togom <math>e</math> kigatang do somok <math>e \approx 2.71828...</math>

==== Tuku sundu ====
<math> \frac{d}{dx}(x^{a}) = a x^{a-1}</math>
==== Tungab pampos tinsait, tintuhun daudau, om tintuhun kiimpou songkukuo ====
<math>\frac{d}{dx}(e^{x}) = e^{x}</math>

<math>\frac{d}{dx}(a^{x}) = a^{x} \ln{a}</math> <math>\quad \textrm{ontok} \quad a > 0 </math>

<math>\frac{d}{dx}(\ln{(x)}) = \frac{1}{x}</math> <math>\quad \textrm{ontok} \quad x > 0 </math>

<math>\frac{d}{dx}(\log_{a}{(x)}) = \frac{1}{x \ln{(a)}}</math> <math>\quad \textrm{ontok} \quad x, a > 0 </math>

==== Tungab pampos tolusunduk ====
<math>\frac{d}{dx}(\sin(x)) = \cos(x) </math>

<math>\frac{d}{dx}(\cos(x)) = - \sin(x) </math>

<math>\frac{d}{dx}(\tan(x)) = \sec^{2}(x) = \frac{1}{1 + \cos^{2}(x)} = 1 + \tan^{2}(x)</math>

==== Tungab pampos bolik tolusunduk ====
<math>\frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1- x^{2}} } </math> <math>\quad \textrm{ontok} \quad x \in (-1,1)</math>

<math>\frac{d}{dx}(\arccos(x)) = - \frac{1}{\sqrt{1- x^{2}} } </math> <math>\quad \textrm{ontok} \quad x \in (-1,1)</math>

<math>\frac{d}{dx}(\arctan(x)) = \frac{1}{1 + x^{2}} </math>

== Sukuon ==