Derivasjonsregler

Definisjon

$f^{\text{'}}\left(x\right)=\underset{∆x\to 0}{\lim }\frac{f(x+∆x)-f\left(x\right)}{∆x}$

Konstant funksjon

$f\left(x\right)=k$

$f^{\text{'}}\left(x\right)=0$

Potensfunksjon

$f\left(x\right)=x^r$

$f^{\text{'}}\left(x\right)=r·x^{r-1}$

Funksjon multiplisert
med konstant

$f\left(x\right)=k·g\left(x\right)$

$$ f^{\text{'}}\left(x\right)=k·g^{\text{'}}\left(x\right)$$

Summer og differanser

$f\left(x\right)=g\left(x\right)\pm h\left(x\right)$

$$ f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right)\pm h^{\text{'}}\left(x\right)$$

Produkt

$f\left(x\right)=u\left(x\right)·v\left(x\right)$

$$ f^{\text{'}}\left(x\right)=u^{\text{'}}\left(x\right)·v\left(x\right)+u\left(x\right)·v^{\text{'}}\left(x\right)$$

Kvotienter (brøk)

$f\left(x\right)=\frac{u\left(x\right)}{v\left(x\right)}$

$$ f^{\text{'}}\left(x\right)=\frac{u^{\text{'}}\left(x\right)·v\left(x\right)-u\left(x\right)·v^{\text{'}}\left(x\right)}{{\left(v\left(x\right)\right)}^2}$$

Eksponentialfunksjoner

$$ f\left(x\right)=e^x$$

$$ f^{\text{'}}\left(x\right)=e^x$$

$f\left(x\right)=a^x$

$f^{\text{'}}\left(x\right)=a^x·\ln a$

Logaritmefunksjoner

$f\left(x\right)=\ln x$

$f^{\text{'}}\left(x\right)=\frac{1}{x}$

Kjerneregelen

$f\left(x\right)=g\left(u\right(x\left)\right)$

$$ f^{\text{'}}\left(x\right)=g^{\text{'}}\left(u\right)·u^{\text{'}}\left(x\right)$$