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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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