Ableitungsregeln

Potenzregel:

$f (x) = x^{n}$

$f^{'} (x) = n \cdot x^{n - 1}$

Beispiel: $f (x) = x^{3}$

$f^{'} (x) = x^{2}$

Faktorregel:

$f (x) = a \cdot g (x)$

$f^{'} (x) = a \cdot g^{'} (x)$

Beispiel: $f (x) = 2 x^{2}$

$f^{'} (x) = 4 x$

Summenregel:

$f (x) = g (x) + h (x)$

$f^{'} (x) = g^{'} (x) + h^{'} (x)$

Beispiel: $f (x) = x^{2} + x^{4}$

$f^{'} (x) = 2 x + 4 x^{3}$

Produktregel:

$f (x) = u (x) \cdot v (x)$

$f^{'} (x) = u^{'} (x) \cdot v (x) + u (x) \cdot v^{'} (x)$

Beispiel: $f (x) = 2 x \cdot (x^{2} + 3 x)$

$f^{'} (x) = 2 \cdot (x^{2} + 3 x) + 2 x \cdot (2 x + 3) = 2 x^{2} + 6 x + 4 x^{2} + 6 x = 6 x^{2} + 12 x$

Quotientenregel:

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

$f (x) = \frac{N \cdot A Z - Z \cdot A N}{N^{2}}$

Beispiel: $f (x) = \frac{2 x - 1}{x + 2}$

$f^{'} (x) = \frac{(x + 2) \cdot 2 - (2 x - 1) \cdot 1}{(x + 2)^{2}} = \frac{2 x + 4 - 2 x + 1}{(x + 2)^{2}} = \frac{4 x + 5}{(x + 2)^{2}}$

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