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Taulukoita

Kreikkalaiset aakkoset

$$\begin{split}\begin{array}{cc | cc}\hline \text{Symboli} & \text{Nimi} & \text{Symboli} & \text{Nimi} \\\hline \alpha \, A & \text{alfa} & \nu \, N & \text{nyy} \\ \beta \, B & \text{beeta} & \xi \, \Xi & \text{ksii} \\ \gamma \, \Gamma & \text{gamma} & o \, O & \text{omikron} \\ \delta \, \Delta & \text{delta} & \pi \, \Pi & \text{pii} \\ \varepsilon \, E & \text{epsilon} & \rho \, P & \text{rhoo} \\ \zeta \, Z & \text{zeeta} & \sigma \, \Sigma & \text{sigma} \\ \eta \, H & \text{eeta} & \tau \, T & \text{tau} \\ \theta \, \Theta & \text{theeta} & \upsilon \, \Upsilon & \text{ypsilon} \\ \iota \, I & \text{ioota} & \phi \, \Phi & \text{fii} \\ \kappa \, K & \text{kappa} & \chi \, X & \text{khii} \\ \lambda \, \Lambda & \text{lambda} & \psi \, \Psi & \text{psii} \\ \mu \, M & \text{myy} & \omega \, \Omega & \text{oomega} \\\hline \end{array}\end{split}$$

Derivointikaavoja

$$\begin{split}\begin{array}{cc|cc|cc}\hline f(x) & f'(x) & f(x) & f'(x) & f(x) & f'(x) \\\hline x^a & ax^{a - 1} & \sin x & \cos x & \sinh x & \cosh x \\[2ex] x^{\frac{1}{a}} & \dfrac{x^{\frac{1}{a} - 1}}{a} & \cos x & -\sin x & \cosh x & \sinh x \\[2ex] e^x & e^x & \tan x & \dfrac{1}{\cos^2 x} & \tanh x & \dfrac{1}{\cosh^2 x} \\[2ex] a^x & a^x\ln a & \arcsin x & \dfrac{1}{\sqrt{1 - x^2}} & \arsinh x & \dfrac{1}{\sqrt{1 + x^2}} \\[2ex] \ln x & \dfrac{1}{x} & \arccos x & -\dfrac{1}{\sqrt{1 - x^2}} & \arcosh x & \dfrac{1}{\sqrt{x^2 - 1}} \\[2ex] \log_a x & \dfrac{1}{x\ln a} & \arctan x & \dfrac{1}{1 + x^2} & \artanh x & \dfrac{1}{1 - x^2} \\[2ex]\hline \end{array}\end{split}$$

$$\begin{split}\begin{array}{cl}\hline \text{Kaava} & \text{Nimi} \\\hline D(cf(x)) = cf'(x) & \text{vakion siirto} \\ D(f(x) \pm g(x)) = f'(x) \pm g'(x) & \text{lineaarisuus} \\ D(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) & \text{tulon derivointi} \\[3ex] D\left(\dfrac{f(x)}{g(x)}\right) = \dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} & \text{osamäärän derivointi} \\[3ex] D((f \circ g)(x)) = f'(g(x))g'(x) & \text{ketjusääntö} \\[3ex] D(f^{-1}(y)) = \dfrac{1}{f'(x)}, \text{ kun } f(x) = y & \text{käänteisfunktion derivointi} \\[3ex]\hline \end{array}\end{split}$$

Perusintegraaleja

$$\begin{split}\begin{array}{ccl}\hline && \\[-3ex] f(x) & \int f(x)\,\d x & \text{Huomioita} \\[1ex]\hline &&\\[-1ex] x^n & \dfrac{x^{n + 1}}{n + 1} + C & n \in \Z \setminus \{-1\}, \text{ ei voimassa pisteen } 0 \text{ yli jos } n < 0 \\[3ex] x^a & \dfrac{x^{a + 1}}{a + 1} + C & a \in \R \setminus \{-1\}, \text{ voimassa kun } x > 0 \\[3ex] \dfrac{1}{x} & \ln|x| + C & \text{ei voimassa pisteen } 0 \text{ yli} \\[3ex] e^x & e^x + C & \\[3ex] \sin x & -\cos x + C & \\[3ex] \cos x & \sin x + C & \\[3ex] \tan x & -\ln\left|\cos x\right| + C & \text{ei voimassa pisteiden } \dfrac{\pi}{2} + n\pi,\ n \in \Z \text{ yli} \\[3ex] \dfrac{1}{\tan x} & \ln\left|\sin x\right| + C & \text{ei voimassa pisteiden } n\pi,\ n \in \Z \text{ yli} \\[3ex] \dfrac{1}{\cos^2 x} & \tan x + C & \text{ei voimassa pisteiden } \dfrac{\pi}{2} + n\pi,\ n \in \Z \text{ yli} \\[3ex] \dfrac{1}{\sin^2 x} & -\dfrac{1}{\tan x} + C & \text{ei voimassa pisteiden } n\pi,\ n \in \Z \text{ yli} \\[3ex] \dfrac{1}{\sqrt{1 - x^2}} & \arcsin x + C & \text{voimassa kun } {-1} < x < 1 \\[3ex] \dfrac{1}{1 + x^2} & \arctan x + C & \\[3ex] \dfrac{1}{\sqrt{1 + x^2}} & \arsinh x + C & \\[3ex] \dfrac{1}{\sqrt{x^2 - 1}} & \arcosh x + C & \text{ei voimassa kun } {-1} < x < 1 \\[3ex] \dfrac{1}{1 - x^2} & \artanh x + C & \text{voimassa kun } {-1} < x < 1 \\[3ex]\hline \end{array}\end{split}$$

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