\[\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\ba}{\mathbf{a}} \newcommand{\bb}{\mathbf{b}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bd}{\mathbf{d}} \newcommand{\be}{\mathbf{e}} \newcommand{\bff}{\mathbf{f}} \newcommand{\bh}{\mathbf{h}} \newcommand{\bi}{\mathbf{i}} \newcommand{\bj}{\mathbf{j}} \newcommand{\bk}{\mathbf{k}} \newcommand{\bN}{\mathbf{N}} \newcommand{\bn}{\mathbf{n}} \newcommand{\bo}{\mathbf{0}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bq}{\mathbf{q}} \newcommand{\br}{\mathbf{r}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bw}{\mathbf{w}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\bzero}{\mathbf{0}} \newcommand{\nv}{\mathbf{0}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\pv}{\overline} \newcommand{\iu}{\mathrm{i}} \newcommand{\ju}{\mathrm{j}} \newcommand{\re}{\operatorname{Re}} \newcommand{\im}{\operatorname{Im}} \newcommand{\arsinh}{\operatorname{ar\,sinh}} \newcommand{\arcosh}{\operatorname{ar\,cosh}} \newcommand{\artanh}{\operatorname{ar\,tanh}} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\proj}{\operatorname{proj}} \newcommand{\rref}{\operatorname{rref}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Span}{\operatorname{span}} \newcommand{\vir}{\operatorname{span}} \renewcommand{\dim}{\operatorname{dim}} \newcommand{\alg}{\operatorname{alg}} \newcommand{\geom}{\operatorname{geom}} \newcommand{\id}{\operatorname{id}} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\tp}[1]{#1^{\top}} \renewcommand{\d}{\mathrm{d}} \newcommand{\sij}[2]{\bigg/_{\mspace{-15mu}#1}^{\,#2}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\pysty}[1]{\left[\begin{array}{@{}r@{}}#1\end{array}\right]} \newcommand{\piste}{\cdot} \newcommand{\qedhere}{} \newcommand{\taumatrix}[1]{\left[\!\!#1\!\!\right]} \newenvironment{augmatrix}[1]{\left[\begin{array}{#1}}{\end{array}\right]} \newenvironment{vaugmatrix}[1]{\left|\begin{array}{#1}}{\end{array}\right|}\]

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¶

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