## Precalculus (6th Edition) Blitzer

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.
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.