Answer
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.
Work Step by Step
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 âfâ 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.
