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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