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.