As the title suggests, I was wondering what could be the geometric meaning of differentials of higher order than the first, being aware that the first order differential is a linear application that approximates linearly the increment of the function itself at a point.
Thanks in advance.
The second derivative of a function corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
Similarly, the third derivative of a function corresponds to the change of the curvature of the graph.