Pochodne podstawowych funkcji elementarnych:
| f(x) | f’(x) |
|---|---|
| const. | 0 |
xa, a ∈ R |
a • xa − 1 |
ax, a > 0 |
ax • lna |
$$\operatorname{}\mathbf{x}\mathbf{,\ a\ > 0\ \hat{}\ a\ \neq 1}$$ |
$$\frac{1}{x \bullet \ln a}$$ |
| sin x | cos x |
| cos x | - sin x |
| tg x | $$\frac{1}{\cos^{2}x}$$ |
| ctg x | - $\frac{1}{\sin^{2}x}$ |
| arc sin x | $$\frac{1}{\sqrt{1 - x^{2}}}$$ |
| arc ctg x | $$\frac{1}{1 + x^{2}}$$ |
(f + g)′(x0)=f′(x0)+g′(x0)
(f − g)′(x0)=f′(x0)−g′(x0)
(c • f)′(x0)=c•f′(x0), dla c∈R
(f • g)′(x0)=f′(x0)•g(x0)+f(x0) • g′(x0)
$$\left( \frac{\mathbf{f}}{\mathbf{g}} \right)^{\mathbf{'}}\left( \mathbf{x}_{\mathbf{0}} \right)\mathbf{=}\frac{\mathbf{f}^{\mathbf{'}}\left( \mathbf{x}_{\mathbf{0}} \right)\mathbf{\bullet g}\left( \mathbf{x}_{\mathbf{0}} \right)\mathbf{- f(}\mathbf{x}_{\mathbf{0}}\mathbf{) \bullet g'(}\mathbf{x}_{\mathbf{0}}\mathbf{)}}{\mathbf{g}^{\mathbf{2}}\mathbf{(x}_{\mathbf{0}}\mathbf{)}}$$