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Taulukoita
Derivointikaavoja
| \(f(x)\) |
\(f'(x)\) |
\(f(x)\) |
\(f'(x)\) |
Kaava |
Nimi |
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| \(x^a\) |
\(ax^{a - 1}\) |
\(x^{\frac{1}{a}}\) |
\(\frac{x^{\frac{1}{a} - 1}}{a}\) |
\((cf)' = cf'\) |
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 |
| \(\sinh x\) |
\(\cosh x\) |
\(\operatorname{ar\,sinh}x\) |
\(\frac{1}{\sqrt{1 + x^2}}\) |
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| \(\cosh x\) |
\(\sinh x\) |
\(\operatorname{ar\,cosh}x\) |
\(\frac{1}{\sqrt{x^2 - 1}}\) |
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| \(\tanh x\) |
\(\frac{1}{\cosh^2 x}\) |
\(\operatorname{ar\,tanh}x\) |
\(\frac{1}{1 - x^2}\) |
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Perusintegraaleja
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| \(f(x)\) |
\(\int f(x)\,\mathrm{d}x\) |
Huomioita |
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| \(x^n\) |
\(\frac{x^{n + 1}}{n + 1} + C\) |
\(n \in \mathbb Z\setminus \{-1\}\), 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 |
| \(e^x\) |
\(e^x + C\) |
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| \(\sin x\) |
\(-\cos x + C\) |
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| \(\cos x\) |
\(\sin x + C\) |
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| \(\tan x\) |
\(-\ln|\cos x| + C\) |
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 \(-1 < x < 1\) |
| \(\frac{1}{1 + x^2}\) |
\(\arctan x + C\) |
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| \(\frac{1}{\sqrt{1 + x^2}}\) |
\(\operatorname{ar\,sinh}x + C\) |
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| \(\frac{1}{\sqrt{x^2 - 1}}\) |
\(\operatorname{ar\,cosh}x + C\) |
ei voimassa kun \(-1 < x < 1\) |
| \(\frac{1}{1 - x^2}\) |
\(\operatorname{ar\,tanh}x + C\) |
voimassa kun \(-1 < x < 1\) |
Sarjakehitelmiä
| Sarjakehitelmä |
Suppenemisväli |
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| \(\frac{1}{1 - x} = \sum_{k = 0}^{\infty}x^k = 1 + x + x^2 + x^3 + x^4 + \cdots\) |
\(-1 < x < 1\) |
| \(e^x = \sum_{k = 0}^{\infty}\frac{x^k}{k!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots\) |
\(\mathbb R\) |
| \(\sin x = \sum_{k = 0}^{\infty}\frac{(-1)^{k}x^{2k + 1}}{(2k + 1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \cdots\) |
\(\mathbb R\) |
| \(\cos x = \sum_{k = 0}^{\infty}\frac{(-1)^kx^{2k}}{(2k)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \frac{x^8}{40320} - \cdots\) |
\(\mathbb R\) |
| \(\ln(1 + x) = \sum_{k = 0}^{\infty}\frac{(-1)^kx^{k + 1}}{k + 1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \cdots\) |
\(-1 < x \leq 1\) |