\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Saturday, June 01, 2002 19:34:59} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - D\U{f4}kazy\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Deriv\'{a}cia funkcie} \subsection{D\^{o}kazy vety o pravidl\'{a}ch derivovania} \textbf{D\^{o}kaz: }Diferencovate\v{l}nos\v{t} funkci\'{\i} $f,g$ v bode $% a\in A$ implikuje, \v{z}e existuj\'{u} vlastn\'{e} limity: \[ f\,\/\,^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a},\,g^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{g\left( x\right) -g\left( a\right) }{x-a}. \]% Pre \textsl{n\'{a}sobok funkcie} m\'{a}me: \[ (cf)^{\prime }(a)=\lim_{x\longrightarrow a}\frac{\left( cf\right) \left( x\right) -\left( cf\right) \left( a\right) }{x-a}=c\lim_{x\longrightarrow a}% \frac{f\left( x\right) -f\left( a\right) }{x-a}=cf\,\/\,^{\prime }\left( a\right) . \]% Pre \textsl{s\'{u}\v{c}et funkci\'{\i}} \[ \left( f+g\right) ^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{% \left( f+g\right) \left( x\right) -\left( f+g\right) \left( a\right) }{x-a}% =\lim_{\longrightarrow }\frac{f\left( x\right) +g\left( x\right) -f\left( a\right) -g\left( a\right) }{x-a}= \]% \[ =\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}% +\lim_{x\longrightarrow a}\frac{g\left( x\right) -g\left( a\right) }{x-a}% =f\,\/\,^{\prime }\left( a\right) +g^{\prime }\left( a\right) . \]% Pre \textsl{s\'{u}\v{c}in funkci\'{\i}} \[ \left( fg\right) ^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{% \left( fg\right) \left( x\right) -\left( fg\right) \left( a\right) }{x-a}% =\lim_{x\longrightarrow a}\frac{f\left( x\right) g\left( x\right) -f\left( a\right) g\left( a\right) +f\left( x\right) g\left( a\right) -f\left( x\right) g\left( a\right) }{x-a}= \]% \[ =g\left( a\right) \lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}+\lim_{x\longrightarrow a}f\left( x\right) \frac{g\left( x\right) -g\left( a\right) }{x-a}=f\,\/\,^{\prime }\left( a\right) g\left( a\right) +\lim_{x\longrightarrow a}f\left( x\right) \lim_{x\longrightarrow a}% \frac{g\left( x\right) -g\left( a\right) }{x-a}= \]% \[ =f\,\/\,^{\prime }\left( a\right) g\left( a\right) +f\left( a\right) g^{\prime }\left( a\right) . \]% V d\^{o}kaze vzorca o deriv\'{a}cii s\'{u}\v{c}inu funkci\'{\i} sme vyu\v{z}% ili vetu, ktor\'{a} hovor\'{\i}, \v{z}e ak je funkcia v bode $a$ diferencovate\v{l}n\'{a}, potom je v tomto bode spojit\'{a}. T\'{u}to vetu vyu\v{z}ijeme aj pri d\^{o}kaze vzorca o deriv\'{a}cii podielu funkci\'{\i}. Ak $g\left( a\right) \neq 0$ a preto\v{z}e $g$ je diferencovate\v{l}n\'{a} v bode $a$, potom je aj spojit\'{a} v bode $a$, teda existuje $O_{\delta }\left( a\right) $ tak\'{e}, \v{z}e $\forall x\in O_{\delta }\left( a\right) \Longrightarrow g\left( x\right) \neq 0$ a pre \textsl{podiel funkci\'{\i}} \[ \left( \frac{f}{g}\right) ^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{\frac{f}{g}\left( x\right) -\frac{f}{g}\left( a\right) }{x-a}% =\lim_{x\longrightarrow a}\frac{\frac{f\left( x\right) }{g\left( x\right) }-% \frac{f\left( a\right) }{g\left( a\right) }}{x-a}=\lim_{x\longrightarrow a}% \frac{f\left( x\right) g\left( a\right) -f\left( a\right) g\left( x\right) +f\left( a\right) g\left( a\right) -f\left( a\right) g\left( a\right) }{% \left( x-a\right) g\left( x\right) g\left( a\right) }= \]% \[ =\lim_{x\longrightarrow a}\frac{\left( f\left( x\right) -f\left( a\right) \right) g\left( a\right) -\left( g\left( x\right) -g\left( a\right) \right) f\left( a\right) }{\left( x-a\right) g\left( x\right) g\left( a\right) }=% \frac{f\,\/\,^{\prime }\left( a\right) g\left( a\right) -f\left( a\right) g^{\prime }\left( a\right) }{g^{2}\left( a\right) }.\,\square \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{M53.tex#1}} \\ \hline \end{tabular} \end{center} \end{document}