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Ableitungsregeln

Mathematischer Grundbegriff
Ableitung eines Vielfachen
Funktion: $f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=c\phantom{\rule{0.12em}{0ex}}\cdot g\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)$
Ableitung :
$f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=c\phantom{\rule{0.12em}{0ex}}\cdot g\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)$

Beispiel: $f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=7\cdot {x\phantom{\rule{0.12em}{0ex}}}^{2}⇒f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=7\cdot 2x\phantom{\rule{0.12em}{0ex}}$

Ableitung einer Summe
Funktion: $h\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)+g\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)$
Ableitung :
$h\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)+g\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)$

Beispiel: $h\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)={x\phantom{\rule{0.12em}{0ex}}}^{2}+{x\phantom{\rule{0.12em}{0ex}}}^{3}⇒h\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=2x\phantom{\rule{0.12em}{0ex}}+3{x\phantom{\rule{0.12em}{0ex}}}^{2}$

Produktregel
Funktion: $f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=u\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)\cdot v\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)$
Ableitung :
$f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=u\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)\cdot v\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)+u\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)\cdot v\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)$

Beispiel: $f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=x\phantom{\rule{0.12em}{0ex}}\cdot sin\left(x\phantom{\rule{0.12em}{0ex}}\right)⇒f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=1\cdot sin\left(x\phantom{\rule{0.12em}{0ex}}\right)+x\phantom{\rule{0.12em}{0ex}}\cdot cos\left(x\phantom{\rule{0.12em}{0ex}}\right)$

Quotientenregel
Funktion: $f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=\frac{u\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)}{v\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)}$
Ableitung :
$f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=\frac{u\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)\cdot v\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)-u\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)\cdot v\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)}{v\phantom{\rule{0.12em}{0ex}}{\left(x\phantom{\rule{0.12em}{0ex}}\right)}^{2}}$

Beispiel: $f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=\frac{2}{x\phantom{\rule{0.12em}{0ex}}}⇒f\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=\frac{0\cdot x\phantom{\rule{0.12em}{0ex}}-2\cdot 1}{{x\phantom{\rule{0.12em}{0ex}}}^{2}}$

Kettenregel
Funktion: $h\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=f\phantom{\rule{0.12em}{0ex}}\left(g\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)\right)$
Ableitung :
$h\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=f\phantom{\rule{0.12em}{0ex}}\text{'}\left(g\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)\right)\cdot g\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)$

Beispiel: $h\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)=sin\left({x\phantom{\rule{0.12em}{0ex}}}^{2}\right)⇒h\phantom{\rule{0.12em}{0ex}}\text{'}\left(x\phantom{\rule{0.12em}{0ex}}\right)=cos\left({x\phantom{\rule{0.12em}{0ex}}}^{2}\right)\cdot 2x\phantom{\rule{0.12em}{0ex}}$

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Kategorie: Ableitung

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