Taulukoita¶
Derivointikaavoja¶
| \(f(x)\) | \(f'(x)\) | \(f(x)\) | \(f'(x)\) | Kaava | Nimi |
|---|---|---|---|---|---|
| \(x^a\) | \(ax^{a - 1}\) | \(x^{\frac{1}{a}}\) | \(\frac{x^{\frac{1}{a} - 1}}{a}\) | \((cf)' = cf'\) | vakion siirto |
| \(e^x\) | \(e^x\) | \(\ln x\) | \(\frac{1}{x}\) | \((f \pm g)' = f' \pm g'\) | lineaarisuus |
| \(a^x\) | \(a^x\ln a\) | \(\log_a x\) | \(\frac{1}{x\ln a}\) | \((fg)' = f'g + fg'\) | tulon derivointi |
| \(\sin x\) | \(\cos x\) | \(\arcsin x\) | \(\frac{1}{\sqrt{1 - x^2}}\) | \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\) | osamäärän derivointi |
| \(\cos x\) | \(-\sin x\) | \(\arccos x\) | \(-\frac{1}{\sqrt{1 - x^2}}\) | \((f \circ g)' = (f' \circ g)g'\) | ketjusääntö |
| \(\tan x\) | \(\frac{1}{\cos^2 x}\) | \(\arctan x\) | \(\frac{1}{1 + x^2}\) | \((f^{-1})' = \frac{1}{(f' \circ f^{-1})}\) | käänteisfunktion derivointi |
| \(\sinh x\) | \(\cosh x\) | \(\operatorname{ar\,sinh}x\) | \(\frac{1}{\sqrt{1 + x^2}}\) | ||
| \(\cosh x\) | \(\sinh x\) | \(\operatorname{ar\,cosh}x\) | \(\frac{1}{\sqrt{x^2 - 1}}\) | ||
| \(\tanh x\) | \(\frac{1}{\cosh^2 x}\) | \(\operatorname{ar\,tanh}x\) | \(\frac{1}{1 - x^2}\) |
Perusintegraaleja¶
| \(f(x)\) | \(\int f(x)\,\mathrm{d}x\) | Huomioita |
| \(x^n\) | \(\frac{x^{n + 1}}{n + 1} + C\) | \(n \in \mathbb Z\setminus \{-1\}\), ei voimassa pisteen \(0\) yli jos \(n < 0\) |
| \(x^a\) | \(\frac{x^{a + 1}}{a + 1} + C\) | \(a \in \mathbb R\setminus \{-1\}\), voimassa kun \(x > 0\) |
| \(\frac{1}{x}\) | \(\ln|x| + C\) | ei voimassa pisteen \(0\) yli |
| \(e^x\) | \(e^x + C\) | |
| \(\sin x\) | \(-\cos x + C\) | |
| \(\cos x\) | \(\sin x + C\) | |
| \(\tan x\) | \(-\ln|\cos x| + C\) | ei voimassa pisteiden \(\frac{\pi}{2} + n\pi\), \(n \in \mathbb Z\) yli |
| \(\frac{1}{\tan x}\) | \(\ln|\sin x| + C\) | ei voimassa pisteiden \(n\pi\), \(n \in \mathbb Z\) yli |
| \(\frac{1}{\cos^2 x}\) | \(\tan x + C\) | ei voimassa pisteiden \(\frac{\pi}{2} + n\pi\), \(n \in \mathbb Z\) yli |
| \(\frac{1}{\sin^2 x}\) | \(-\frac{1}{\tan x} + C\) | ei voimassa pisteiden \(n\pi\), \(n \in \mathbb Z\) yli |
| \(\frac{1}{\sqrt{1 - x^2}}\) | \(\arcsin x + C\) | voimassa kun \(-1 < x < 1\) |
| \(\frac{1}{1 + x^2}\) | \(\arctan x + C\) | |
| \(\frac{1}{\sqrt{1 + x^2}}\) | \(\operatorname{ar\,sinh}x + C\) | |
| \(\frac{1}{\sqrt{x^2 - 1}}\) | \(\operatorname{ar\,cosh}x + C\) | ei voimassa kun \(-1 < x < 1\) |
| \(\frac{1}{1 - x^2}\) | \(\operatorname{ar\,tanh}x + C\) | voimassa kun \(-1 < x < 1\) |
Sarjakehitelmiä¶
| Sarjakehitelmä | Suppenemisväli |
|---|---|
| \(\frac{1}{1 - x} = \sum_{k = 0}^{\infty}x^k = 1 + x + x^2 + x^3 + x^4 + \cdots\) | \(-1 < x < 1\) |
| \(e^x = \sum_{k = 0}^{\infty}\frac{x^k}{k!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots\) | \(\mathbb R\) |
| \(\sin x = \sum_{k = 0}^{\infty}\frac{(-1)^{k}x^{2k + 1}}{(2k + 1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \cdots\) | \(\mathbb R\) |
| \(\cos x = \sum_{k = 0}^{\infty}\frac{(-1)^kx^{2k}}{(2k)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \frac{x^8}{40320} - \cdots\) | \(\mathbb R\) |
| \(\ln(1 + x) = \sum_{k = 0}^{\infty}\frac{(-1)^kx^{k + 1}}{k + 1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \cdots\) | \(-1 < x \leq 1\) |