Answer
The slope of the graph of a function at a point $\left( a,f\left( a \right) \right)$ gives the instantaneous rate of change of $f$ with respect to x at a.
Work Step by Step
The slope of the graph of a function at a point $\left( a,f\left( a \right) \right)$ is given by,
${{m}_{\tan }}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$
For example: Consider a function $f\left( x \right)=x+4$ and the point $\left( 1,5 \right)$.
$\begin{align}
& f\left( a+h \right)=1+h+4 \\
& =5+h
\end{align}$
$\begin{align}
& f\left( a \right)=1+4 \\
& =5
\end{align}$
Substitute these values in ${{m}_{\tan }}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$.
$\begin{align}
& {{m}_{\tan }}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h} \\
& =\underset{h\to 0}{\mathop{\lim }}\,\frac{5+h-5}{h} \\
& =\underset{h\to 0}{\mathop{\lim }}\,\frac{h}{h} \\
& =1
\end{align}$
Hence, the slope of the equation $f\left( x \right)=x+4$ at $\left( 1,5 \right)$ is 1. It gives instantaneous rate of change of $f$ with respect to x at a.
