Any function usually has a slope at any point of its curve that is given by the derivative of that point.
You can now also make a graph of the values of these slopes (derivatives). Here you
see the curve of a function (in red) together with the corresponding graph (in blue), the graph given by .
The function is called the slope function or derivative function.
If you compare the two curves you can see that:
the values of the slope function are positive while the original function is increasing;
the values of the slope function are negative while the original function is decreasing;
at values of where the slope function has a value of , the original function has a horizontal tangent; these are often extrema of the original function. .
It is therefore primarily the sign (positive, negative or ) of the derivative function that provides information about the curve of the original function.