# Tabela odvodov

Odvod je osnovna operacija infinitezimalnega računa. Spodaj so navedena najpomembnejša pravila za odvajanje.

### Pravila za sestavljanje funkcije

funkcija odvod opombe
${\displaystyle f\pm g}$ ${\displaystyle f'\pm g'}$
${\displaystyle c\cdot f}$ ${\displaystyle c\cdot f'}$ c je konstanta
${\displaystyle f\cdot g}$ ${\displaystyle f'\cdot g+f\cdot g'}$ odvod produkta
${\displaystyle {f \over g}}$ ${\displaystyle {{f'\cdot g-f\cdot g'} \over g^{2}}}$ ${\displaystyle g\neq 0}$
odvod kvocienta
${\displaystyle f\circ g=f(g)}$ ${\displaystyle (f^{\prime }\circ g)\cdot g'=f'(g)\cdot g'}$

### Odvodi elementarnih funkcij

Funkcija Odvod Opombe
${\displaystyle c\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle c\in \mathbb {R} }$
${\displaystyle x\!\,}$ ${\displaystyle 1\!\,}$
${\displaystyle x^{n}\!\,}$ ${\displaystyle nx^{n-1}\!\,}$ ${\displaystyle n\in \mathbb {R} }$
${\displaystyle 1 \over x}$ ${\displaystyle -{1 \over x^{2}}}$ ${\displaystyle x\neq 0\!\,}$
${\displaystyle 1 \over x^{2}}$ ${\displaystyle -{2 \over x^{3}}}$ ${\displaystyle x\neq 0\!\,}$
${\displaystyle {\sqrt {x}}}$ ${\displaystyle 1 \over 2{\sqrt {x}}}$ ${\displaystyle x>0\!\,}$
${\displaystyle {\sqrt[{n}]{x}}}$ ${\displaystyle 1 \over n{\sqrt[{n}]{x^{n-1}}}}$ ${\displaystyle x>0\!\,}$
${\displaystyle \sin x\!\,}$ ${\displaystyle \cos x\!\,}$
${\displaystyle \sin(ax)}$ ${\displaystyle a\cos(ax)}$
${\displaystyle \cos x\!\,}$ ${\displaystyle -\sin x\!\,}$
${\displaystyle \cos(ax)}$ ${\displaystyle -a\sin(ax)}$
${\displaystyle \tan x}$ ${\displaystyle 1 \over \cos ^{2}x}$ ${\displaystyle x\neq {\pi \over 2}+k\pi ,\;k\in \mathbb {Z} }$
${\displaystyle \cot x}$ ${\displaystyle -{1 \over \sin ^{2}x}}$ ${\displaystyle x\neq k\pi ,\;k\in \mathbb {Z} }$
${\displaystyle e^{x}\!\,}$ ${\displaystyle e^{x}\!\,}$
${\displaystyle e^{k}\!\,^{x}}$ ${\displaystyle ke^{k}\!\,^{x}}$
${\displaystyle a^{x}\!\,}$ ${\displaystyle a^{x}\ln a\!\,}$ ${\displaystyle a>0\!\,}$
${\displaystyle x^{x}\!\,}$ ${\displaystyle x^{x}(1+\ln x)\!\,}$ ${\displaystyle x>0\!\,}$
${\displaystyle \ln x\!\,}$ ${\displaystyle 1 \over x}$ ${\displaystyle x>0\!\,}$
${\displaystyle \log _{a}x\!\,}$ ${\displaystyle 1 \over x\ln a}$ ${\displaystyle a>0,~~x>0\!\,}$
${\displaystyle \arcsin x}$ ${\displaystyle 1 \over {\sqrt {1-x^{2}}}}$ ${\displaystyle |x|<1\!\,}$
${\displaystyle \arccos x}$ ${\displaystyle -{1 \over {\sqrt {1-x^{2}}}}}$ ${\displaystyle |x|<1\!\,}$
${\displaystyle \arctan x}$ ${\displaystyle 1 \over 1+x^{2}}$
${\displaystyle \operatorname {arccot} x}$ ${\displaystyle -{1 \over 1+x^{2}}}$
${\displaystyle \operatorname {sh} x={{e^{x}-e^{-x}} \over 2}}$ ${\displaystyle \operatorname {ch} x={{e^{x}+e^{-x}} \over 2}}$
${\displaystyle \operatorname {ch} x={{e^{x}+e^{-x}} \over 2}}$ ${\displaystyle \operatorname {sh} x={{e^{x}-e^{-x}} \over 2}}$
${\displaystyle \operatorname {th} x={\operatorname {sh} x \over \operatorname {ch} x}}$ ${\displaystyle {1 \over \operatorname {ch} ^{2}x}={4 \over ({e^{x}+e^{-x}})^{2}}}$
${\displaystyle \operatorname {arth} x={1 \over 2}\ln {1+x \over 1-x}}$ ${\displaystyle 1 \over 1-x^{2}}$ ${\displaystyle |x|<1\!\,}$
${\displaystyle \ln(x+{\sqrt {x^{2}\pm a^{2}}})}$ ${\displaystyle 1 \over {\sqrt {x^{2}\pm a^{2}}}}$