## Precalculus (6th Edition) Blitzer

The meaning is that the slope of a function at a point is the derivative of the function $f$ 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 x 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. In particular, at $x=a$, the derivative is given by, $f'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$ The values of the derivative of “f” at $x=a$ and the slope of the tangent line to the graph of a function at $\left( a,f\left( a \right) \right)$ are both equal. Thus, the slope of a function at a point is the derivative of the function $f$ at that point.