Rregullat e derivimit

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– Posted on 09/08/2013Posted in: USHTRIME MATEMATIKE

 

\left ( f+g \right )'=f'+g'

\left ( f\cdot g \right )'=f'\cdot g+f\cdot g'

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

\left ( f^{n} \right )'=n\cdot f^{n-1}\cdot f'

\left ( c\cdot f \right )'=c\cdot 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 u'_{x}

Tabela e derivateve

\left ( c \right )'=0

\left ( u^{\alpha } \right )'=\alpha \cdot u^{\alpha -1}\cdot u'

\left ( a^{u} \right )'=a^{u}\cdot lna\cdot u'

\left ( e^{u} \right )'=e^{u}\cdot u'

\left ( log_{a}u \right )'=\frac{1}{u\cdot lna}\cdot u'

\left ( lnu \right )'=\frac{1}{u}\cdot u'

   \left ( sinu \right )'=cosu\cdot u'

\left ( cosu \right )'=-sinu\cdot u'

\left ( tgu \right )'=\frac{1}{cos^{2}u}\cdot u'

\left ( cotgu \right )'=-\frac{1}{sin^{2}u}\cdot 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 ( 21 \right )'=0

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

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

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

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

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