Question

What characteristics of the graph of a function can we discuss by using the concept of differentiation (first and second derivatives).

Answer 1

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The first derivative of a function describes the rate of change of the function. The rate of change is the same as the slope.
The second derivative gives the rate of change OF the rate of change, so I might be increasing, but slowing down, or I might be decreasing but speeding up. As far as what that looks like on a graph, it's a feature called concavity. If it's bent up like a "U" then it's called concave up, and that's positive second derivative. If it's bent down like an "n," then it's called concave down and that's negative second derivative.