# Rregullat e derivimit

$\left&space;(&space;f+g&space;\right&space;)'=f'+g'$

$\left&space;(&space;f\cdot&space;g&space;\right&space;)'=f'\cdot&space;g+f\cdot&space;g'$

$\left&space;(&space;\frac{f}{g}&space;\right&space;)'=\frac{f'\cdot&space;g-f\cdot&space;g'}{g^{2}}$

$\left&space;(&space;f^{n}&space;\right&space;)'=n\cdot&space;f^{n-1}\cdot&space;f'$

$\left&space;(&space;c\cdot&space;f&space;\right&space;)'=c\cdot&space;f'$

Për funksionin y=f[u(x)] përbërje e dy  funksioneve f: y=f(u) dhe u: x→u(x)

$y'_{x}=y'_{u}\cdot&space;u'_{x}$

### Tabela e derivateve

 $\left&space;(&space;c&space;\right&space;)'=0$ $\left&space;(&space;u^{\alpha&space;}&space;\right&space;)'=\alpha&space;\cdot&space;u^{\alpha&space;-1}\cdot&space;u'$ $\left&space;(&space;a^{u}&space;\right&space;)'=a^{u}\cdot&space;lna\cdot&space;u'$ $\left&space;(&space;e^{u}&space;\right&space;)'=e^{u}\cdot&space;u'$ $\left&space;(&space;log_{a}u&space;\right&space;)'=\frac{1}{u\cdot&space;lna}\cdot&space;u'$ $\left&space;(&space;lnu&space;\right&space;)'=\frac{1}{u}\cdot&space;u'$ $\left&space;(&space;sinu&space;\right&space;)'=cosu\cdot&space;u'$ $\left&space;(&space;cosu&space;\right&space;)'=-sinu\cdot&space;u'$ $\left&space;(&space;tgu&space;\right&space;)'=\frac{1}{cos^{2}u}\cdot&space;u'$ $\left&space;(&space;cotgu&space;\right&space;)'=-\frac{1}{sin^{2}u}\cdot&space;u'$

Tabela është për funksionin e përbërë y=f[u(x)]  ku u: x→u(x) është i derivueshëm në pikën x.

### Shembuj:

1.$\left&space;(&space;21&space;\right&space;)'=0$

2.$\left&space;[\left&space;(&space;4x^{2}&space;-2x\right&space;)^{5}&space;\right&space;]'=5\cdot&space;\left&space;(&space;4x^{2}-2x&space;\right&space;)^{4}\cdot&space;\left&space;(&space;4x^{2}&space;-2x\right&space;)'=5\cdot&space;\left&space;(&space;4x^{2}-2x&space;\right&space;)^{4}\cdot&space;\left&space;(&space;8x-2&space;\right&space;)=10\cdot&space;\left&space;(&space;4x^{2}-2x&space;\right&space;)^{4}\cdot&space;\left&space;(&space;4x-1&space;\right&space;)$

Derivati i funksionit $y=u^{5}$  ku  $u(x)=4x^{2}-2x$

3.$\left&space;(2^{2x+5}&space;\right&space;)'=2^{2x+5}\cdot&space;ln2\cdot&space;\left&space;(&space;2x+5&space;\right&space;)'=2\cdot&space;2^{2x+5}\cdot&space;ln2$

4.$\left&space;[&space;ln\left&space;(&space;x^{2}+2&space;\right&space;)+sin\left&space;(&space;5x-e^{x}&space;\right&space;)&space;\right&space;]'=\left&space;[&space;ln\left&space;(&space;x^{2}+2&space;\right&space;)&space;\right&space;]'+\left&space;[&space;sin\left&space;(&space;5x-e^{x}&space;\right&space;)&space;\right&space;]'=\frac{1}{x^{2}+2}\cdot&space;\left&space;(&space;x^{2}+2&space;\right&space;)'+cos\left&space;(&space;5x-e^{x}&space;\right&space;)\cdot&space;\left&space;(&space;5x-e^{x}&space;\right&space;)'=\frac{2x}{x^{2}+2}+\left&space;(&space;5-e^{x}\&space;\right&space;)\cdot&space;cos\left&space;(&space;5x-e^{x}&space;\right&space;)$