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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Derivatives">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 18:39:42 1997}
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\section{Derivatives\label{Derivative}}

\label{Differential calculus}Differential calculus is concerned with how one
quantity changes in relation to another quantity. The central concept of
differential calculus is the derivative.

\subsubsection{Derivative}

\begin{definition}
The \textbf{derivative of a function $f$ at a number }$a$, denoted by $%
f^{\prime }(a)$, is 
\[
f^{\prime }(a)=\lim_{h\rightarrow 0}\,\,\frac{f(a+h)-f(a)}{h}
\]
if this limit exists.
\end{definition}

\subsubsection{\label{Equation of tangent line}Equation of Tangent Line}

\begin{theorem}
If $f^{\prime }(a)$ exists, then an equation of the tangent line to the
curve $y=f(x)$ at the point $(a,f(a))$ is 
\[
y-f(a)=f^{\prime }(a)(x-a)
\]
\end{theorem}

\subsubsection{\label{Differentiable}Differentiable}

\begin{definition}
A function $f$ is \textbf{differentiable at }$a$ if $f^{\prime }(a)$ exists.
It is \textbf{differentiable on an open interval} $(a,b)$ \lbrack or $%
(a,\infty )\,$or $(-\infty ,a)$ or\thinspace $(-\infty ,\infty )$\rbrack\ if
it is differentiable at every number in the interval.
\end{definition}

\subsubsection{\label{Differentiable implies continuous}Differentiable
Implies Continuous}

\begin{theorem}
If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
\end{theorem}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Computing a derivative}{}{}{../../Help/DM7-3.tex#Derivative%
}}

{\small \hyperref{Dereviatives of piecewise-defined functions}{}{}{%
../../Help/DM7-3.tex#Piecewise-defined function}}

{\small \hyperref{Plotting derivatives}{}{}{../../Help/DM7-3.tex#Plotting
Derivatives}}

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