While the tangent line is a very useful tool, when it comes to investigate the graph of a function, the tangent line fails to say anything about how the graph of a function "bends" at a point. This is where the second derivative comes into play.
Example. Consider the function f(x) = ax^{2}. The tangent line at 0 is the xaxis regardless of the value of a. But if we change a, the graph of f(x) bends more or less sharply depending on the size of the parameter a:
Definition. Let f(x) be a differentiable function on an interval I.
 (i)
 We will say that the graph of f(x) is concave up on I iff f '(x) is increasing on I.
 (ii)
 We will say that the graph of f(x) is concave down on I iff f '(x) is decreasing on I.
Let f(x) be a differentiable function on an interval I. Assume that f '(x) is also differentiable on I.

Example. Consider the function f(x) = x^{9/5}  x. This function is continuous and differentiable for all x. We have
Clearly f ''(0) does not exist. In fact, f '(x) has a vertical tangent at 0. More precisely we have for
which implies
 (1)
 f ''(x) > 0 for x > 0;
 (2)
 f ''(x) < 0 for x < 0.