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Let the real function  be defined and differentiable on the open interval  . Then for every  , there exists the value  as a certain real number. This means that we have a new
function
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(1) |
the so-called derivative function of  ; it is denoted by
or simply  .
Forming the derivative function of a function is called differentiation, the corresponding verb is differentiate.
If the derivative function  is differentiable on  , then we have again a new function, the derivative function of the derivative function of  , which is denoted by  . Formally,
The function
 is callet the second order derivative or the second derivative of  . Similarly, one can call (1) the first (order) derivative of  .
Example. The first derivative of
 is
 and the second derivative is
 , since
If also  is a differentiable function, its derivative function is denoted by  and called the third (order) derivative of  , and so on.
Generally,  can have the derivatives of first, second, third, ...,  th order, where  may be an arbitrarily big positive integer. If  is four or greater, the  th derivative of
 is usually denoted by  . In addition, it's sometimes convenient to think that the 0th order derivative  of  is the function  itself.
The phrase “ is infinitely derivable” means that  has the derivatives of all orders.
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