我不知道如何修复不需要的空格。这只出现在文本的某些位置。我的序言
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%--------this is to get chapter name as small caps and sansserif
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%--------------------------------------------------------------------------------------------
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% **main file**
% \input{preamble}
\usepackage{graphicx} % Required for inserting images
%\input{macros}
\title{sandbox}
\author{latex noob}
\date{\today}
\begin{document}
\frenchspacing
\maketitle
\frontmatter %Use lowercase Roman numerals for page numbers
\tableofcontents
\clearpage
\mainmatter
\chapter{The Derivative}
\section{Introduction}
Consider a quantity \(y=f(x)\) whose value changes with \(x\). Let \(y_0=f(x_0)\) be the value of \(f(x)\) at \(x=x_0\) and \(y_{x_0+\Delta x}=f(x_0+\Delta x)\) at \(x_0+\Delta x\). The difference \(y_{x_0+\Delta x}-y_0=f(x_0+\Delta x)-f(x_0)\) indicates how \(f(x)\) has changed (increment or decrement) in the interval \((x_0,\;x_0+\Delta x)\) and the ratio
\begin{align}\label{eq:1}
\dfrac{f(x_0+\Delta x)-f(x_0)}{x_0+\Delta x-x_0}=\dfrac{\Delta f}{\Delta x},
\end{align}
shows how much \(f(x)\) has changed (increased or decreased) on average per unit change in \(x\) in the interval \((x_0,\;x_0+\Delta x)\). That is, the ratio represents the {\it average rate of change of \(f(x)\)}. Now, The most interesting question we can ask ourselves is how much did \(f(x)\) actually change at the point \(x_0\) alone?
The answer to the above question can be obtained by making \(\Delta x\) progressively smaller, that is, in the limit \(\Delta x\to 0\), then (\ref{eq:1}) yields the {\it (instantaneous) rate of change of \(f(x)\)} at \(x_0\). Mathematically,
\begin{align}\label{eq:2}
\lim\limits_{\Delta x\to 0}\;\dfrac{f\left(x_0+\Delta x\right)-f(x_0)}{\left(x_0+\Delta x\right)-x_0}.
\end{align}
Equation (\ref{eq:2}) is known as the {\it derivative} of \(f(x)\) with respect to \(x\) and is denoted by \(\dfrac{df\left(x\right)}{dx}\) or \(\dfrac{df}{dx}\) or \(f^{\prime}\).\footnote{As \(y=f(x)\), \(\dfrac{dy}{dx}\) or \(y^{\prime}\) are often in literature to represent the derivative of \(y\) with respect to \(x\).}Therefore,
\begin{align}\label{eq:3}
\dfrac{df}{dx}= \lim\limits_{\Delta x\to 0}\;\dfrac{f\left(x_0+\Delta x\right)-f(x_0)}{\left(x_0+\Delta x\right)-x_0}.
\end{align}
\end{document}
