## Precalculus (6th Edition) Blitzer

The derivative of the function $f$ at any point gives the slope of the tangent line to the graph of the function at that point.
The slope of the tangent line to the graph of a function at a point $\left( a,f\left( a \right) \right)$ is given by ${{m}_{\tan }}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$ The derivative of “f” at $\left( x,f\left( x \right) \right)$ is given by $f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( x+h \right)-f\left( x \right)}{h}$ provided this limit exists. The slope of a tangent line can also be found at $\left( x,f\left( x \right) \right)$ where $x$ can represent any number in the domain of $f$ for which the slope of the function is defined. The resulting function would be the derivative of $f$ at $x$. Thus, the derivative of the function $f$ at any point gives the slope of the tangent line to the graph of the function at that point.