\[\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{\rA}{\mathrm{A}} \newcommand{\rB}{\mathrm{B}} \newcommand{\rC}{\mathrm{C}} \newcommand{\rD}{\mathrm{D}} \newcommand{\rE}{\mathrm{E}} \newcommand{\rF}{\mathrm{F}} \newcommand{\rG}{\mathrm{G}} \newcommand{\rH}{\mathrm{H}} \newcommand{\rI}{\mathrm{I}} \newcommand{\rJ}{\mathrm{J}} \newcommand{\rK}{\mathrm{K}} \newcommand{\rL}{\mathrm{L}} \newcommand{\rM}{\mathrm{M}} \newcommand{\rN}{\mathrm{N}} \newcommand{\rO}{\mathrm{O}} \newcommand{\rP}{\mathrm{P}} \newcommand{\rQ}{\mathrm{Q}} \newcommand{\rR}{\mathrm{R}} \newcommand{\rS}{\mathrm{S}} \newcommand{\rT}{\mathrm{T}} \newcommand{\rU}{\mathrm{U}} \newcommand{\rV}{\mathrm{V}} \newcommand{\rW}{\mathrm{W}} \newcommand{\rX}{\mathrm{X}} \newcommand{\rY}{\mathrm{Y}} \newcommand{\rZ}{\mathrm{Z}} \newcommand{\pv}{\overline} \newcommand{\iu}{\mathrm{i}} \newcommand{\ju}{\mathrm{j}} \newcommand{\im}{\mathrm{i}} \newcommand{\e}{\mathrm{e}} \newcommand{\real}{\operatorname{Re}} \newcommand{\imag}{\operatorname{Im}} \newcommand{\Arg}{\operatorname{Arg}} \newcommand{\Ln}{\operatorname{Ln}} \DeclareMathOperator*{\res}{res} \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|} \newcommand{\trans}{\mathrm{T}} \newcommand{\EUR}{\text{\unicode{0x20AC}}} \newcommand{\SI}[3][]{#2\,\mathrm{#3}} \newcommand{\si}[2][]{\mathrm{#2}} \newcommand{\num}[2][]{#2} \newcommand{\ang}[2][]{#2^{\circ}} \newcommand{\meter}{m} \newcommand{\metre}{\meter} \newcommand{\kilo}{k} \newcommand{\kilogram}{kg} \newcommand{\gram}{g} \newcommand{\squared}{^2} \newcommand{\cubed}{^3} \newcommand{\minute}{min} \newcommand{\hour}{h} \newcommand{\second}{s} \newcommand{\degreeCelsius}{^{\circ}C} \newcommand{\per}{/} \newcommand{\centi}{c} \newcommand{\milli}{m} \newcommand{\deci}{d} \newcommand{\percent}{\%} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\Tasd}{\operatorname{Tasd}} \newcommand{\Ber}{\operatorname{Ber}} \newcommand{\Bin}{\operatorname{Bin}} \newcommand{\Geom}{\operatorname{Geom}} \newcommand{\Poi}{\operatorname{Poi}} \newcommand{\Hyperg}{\operatorname{Hyperg}} \newcommand{\Tas}{\operatorname{Tas}} \newcommand{\Exp}{\operatorname{Exp}} \newcommand{\tdist}{\operatorname{t}} \newcommand{\rd}{\mathrm{d}}\]

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

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