What characteristics of the graph of a function can we discuss by using the concept of differentiation (first and second derivatives).
(1) The first derivative can be used to tell the intervals where a function is increasing or decreasing, as the first derivative is positive when the function is increasing and negative where the function is decreasing.
Also, extrema (local maximums or minimums) can be found using the first derivative. Extrema only occur at critical points, which are found when the first derivative is zero or fails to exist. If a function's first derivative is positive for x<c, zero when x=c, and negative when x>c then the function has a local maximum at x=c, etc...
The first derivative gives the slope of the tangent line drawn to the curve at a given point.
(2) The second derivative can be used to find the intervals where the graph is concave up or concave down. This tells you how the rate of change is changing.
Along with the x and y intercepts, vertical asymptotes, and horizontal asymptotes you can get a good feel in order to draw the sketch of a graph.