The Derivative

The Second Derivative

Given a function f , we have defined its derivative f' . This allows us to now define f'' , the second derivative of f , to be the derivative of f' . In terms of the graph of f , the second derivative tells how quickly the slope of the graph is changing near a particular point. Alternately, the second derivative tells how quickly the rate of change of a function is changing.

Since f'' is again a function, we can define f''' = f ⁽³⁾ to be the derivative of f'' . Continuing in this way, we can define the nth derivative of f , f ⁽ⁿ⁾ . The derivative of a differentiable function need not be differentiable everywhere (or even anywhere), so in general the domains of the successive derivatives f ⁽ n) shrink as n increases.