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Taulukoita

Derivointikaavoja

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

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

Perusintegraaleja

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

Sarjakehitelmiä

$$\begin{split}\begin{array}{ll}\hline \text{Sarjakehitelmä} & \text{Suppenemisväli} \\\hline & \\[-1ex] \displaystyle\dfrac{1}{1 - x} = \sum_{k = 0}^{\infty}x^k = 1 + x + x^2 + x^3 + x^4 + \cdots & -1 < x < 1 \\[3ex] \displaystyle e^x = \sum_{k = 0}^{\infty}\dfrac{x^k}{k!} = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \dfrac{x^4}{24} + \cdots & \R \\[3ex] \displaystyle \sin x = \sum_{k = 0}^{\infty}\dfrac{(-1)^{k}x^{2k + 1}}{(2k + 1)!} = x - \dfrac{x^3}{6} + \dfrac{x^5}{120} - \dfrac{x^7}{5040} + \dfrac{x^9}{362880} - \cdots & \R \\[3ex] \displaystyle\cos x = \sum_{k = 0}^{\infty}\dfrac{(-1)^kx^{2k}}{(2k)!} = 1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} - \dfrac{x^6}{720} + \dfrac{x^8}{40320} - \cdots & \R \\[3ex] \displaystyle\ln(1 + x) = \sum_{k = 0}^{\infty}\dfrac{(-1)^kx^{k + 1}}{k + 1} = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \dfrac{x^5}{5} - \cdots & -1 < x \leq 1 \\[3ex]\hline \end{array}\end{split}$$

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