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.
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