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- MAT-01330
- 8. Liitteet
- 8.1 Taulukoita
Taulukoita¶
Derivointikaavoja¶
\(f(x)\) | \(f'(x)\) | \(f(x)\) | \(f'(x)\) | \(f(x)\) | \(f'(x)\) |
---|---|---|---|---|---|
\(x^a\) | \(ax^{a - 1}\) | \(\sin x\) | \(\cos x\) | \(\sinh x\) | \(\cosh x\) |
\(x^{\frac{1}{a}}\) | \(\frac{x^{\frac{1}{a} - 1}}{a}\) | \(\cos x\) | \(-\sin x\) | \(\cosh x\) | \(\sinh x\) |
\(e^x\) | \(e^x\) | \(\tan x\) | \(\frac{1}{\cos^2 x}\) | \(\tanh x\) | \(\frac{1}{\cosh^2 x}\) |
\(a^x\) | \(a^x\ln a\) | \(\arcsin x\) | \(\frac{1}{\sqrt{1 - x^2}}\) | \(\operatorname{ar\,sinh}x\) | \(\frac{1}{\sqrt{1 + x^2}}\) |
\(\ln x\) | \(\frac{1}{x}\) | \(\arccos x\) | \(-\frac{1}{\sqrt{1 - x^2}}\) | \(\operatorname{ar\,cosh}x\) | \(\frac{1}{\sqrt{x^2 - 1}}\) |
\(\log_a x\) | \(\frac{1}{x\ln a}\) | \(\arctan x\) | \(\frac{1}{1 + x^2}\) | \(\operatorname{ar\,tanh}x\) | \(\frac{1}{1 - x^2}\) |
Kaava | Nimi |
---|---|
\(D(cf(x)) = cf'(x)\) | vakion siirto |
\(D(f(x) \pm g(x)) = f'(x) \pm g'(x)\) | lineaarisuus |
\(D(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)\) | tulon derivointi |
\(D\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\) | osamäärän derivointi |
\(D((f \circ g)(x)) = f'(g(x))g'(x)\) | ketjusääntö |
\(D(f^{-1}(y)) = \frac{1}{f'(x)}\), kun \(f(x) = y\) | käänteisfunktion derivointi |
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 |
---|---|
\(\frac{1}{1 - x} = \sum_{k = 0}^{\infty}x^k = 1 + x + x^2 + x^3 + x^4 + \cdots\) | \(-1 < x < 1\) |
\(e^x = \sum_{k = 0}^{\infty}\frac{x^k}{k!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots\) | \(\mathbb R\) |
\(\sin x = \sum_{k = 0}^{\infty}\frac{(-1)^{k}x^{2k + 1}}{(2k + 1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \cdots\) | \(\mathbb R\) |
\(\cos x = \sum_{k = 0}^{\infty}\frac{(-1)^kx^{2k}}{(2k)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \frac{x^8}{40320} - \cdots\) | \(\mathbb R\) |
\(\ln(1 + x) = \sum_{k = 0}^{\infty}\frac{(-1)^kx^{k + 1}}{k + 1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \cdots\) | \(-1 < x \leq 1\) |
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