(f+g)′ = f′ + g′
(f*g)′ = f′g = fg′
$$\left( \frac{f}{g} \right)^{'} = \frac{f^{'}g - fg"}{g^{2}}$$
(cf)′ = cf′; c − stala
(xn)′ = nxn − 1
(ax)′ = (lna)ax
(ex)’=ex
$$\left( \text{lnx} \right)^{'} = \frac{1}{x}$$
(cosx)′ = −sinx
(sinx)′ = cosx
$$\left( \text{tgx} \right)^{'} = \frac{1}{\cos^{2}x}$$
$$\left( \text{arcsinx} \right)^{'} = \frac{1}{\sqrt{1 - x^{2}}}$$
$$\left( \arccos x \right)^{'} = - \frac{1}{\sqrt{1 - x^{2}}}$$
$$\left( \text{arctgx} \right)^{'} = \frac{1}{1 + x^{2}}$$
Styczna do wykresu:
y − f(x0) = f′(x0)(x − x0)
Przybliżenia:
f(y) ≈ f(x0) = f′(x0)x
x = y − x0
Taylor:
$$\text{Pn}\left( x \right) = f\left( x_{0} \right) + \frac{f^{'}\left( x_{0} \right)}{1!}\left( x - x_{0} \right) + \frac{f^{''}\left( x_{0} \right)}{2!}\left( x - x_{0} \right)^{2} + \frac{f^{'''}\left( x_{0} \right)}{3!}\left( x - x_{0} \right)^{3} + \ldots + \frac{f^{n}\left( x_{0} \right)}{n!}\left( x - x_{0} \right)^{n}$$
d’Hospital:
$$\operatorname{}{\left( \frac{f(x)}{g(x)} \right) =}\operatorname{}\left( \frac{f'(x)}{g'(x)} \right)$$
$$\operatorname{}{f\left( x \right)g\left( x \right) = \lim_{x \rightarrow x_{0}}\frac{f(x)}{(\frac{1}{g\left( x \right)})} = \lim_{x \rightarrow x_{0}}\frac{f'(x)}{(\frac{1}{g\left( x \right)})'}} = \lim_{x \rightarrow x_{0}}\frac{g'(x)}{(\frac{1}{f\left( x \right)})'}$$
Kąt między wykresami:
$$tg \propto = \left| \frac{f^{'}\left( x_{0} \right) - g^{'}(x_{0})}{1 + f^{'}\left( x_{0} \right)g^{'}(x_{0})} \right|$$
Błąd pomiaru wielkości f:
f = |f′(x0)|x
Zamiana na e:
limx → x0f(x)g(x) = [1∞] = limx → x0eg(x)lnf(x)
Na przedziale (a,b):
f′(x) > 0 f.rosnaca
f′(x) < 0 f.malejaca
f′(x) = 0 i f″(x)>0 min.lokalne
f′(x) = 0 i f″(x)<0 max.lokalne
f″(x) > 0 wypukla∪
f″(x) < 0 wklesla∩
Badanie funkcji:
1.Dziedzina funkcji
2.Asymptoty pionowe, granice w ± ∞
3.Znak f’(x), monotoniczność
4.Znak f’’(x), wypukłość
5.Tabelka, wykres(szkic)