# Differentiable functions/R/Higher derivatives/Introduction/Section

The derivative of a differentiable function is also called the *first derivative* of . The zeroth derivative is the function itself. Higher derivatives are defined recursively.

Let denote an interval, and let

be a
function.
The function is called -times *differentiable*, if it is -times differentiable, and the -th derivative, that is , is also
differentiable.
The derivative

*derivative*of .

The second derivative is written as , the third derivative as . If a function is -times differentiable, then we say that the derivatives exist up to *order* . A function is called *infinitely often differentiable*, if it is -times differentiable for every .

A differentiable function is continuous due to fact, but its derivative is not necessarily so. Therefore, the following concept is justified.

Let be an interval, and let

be a
function.
The function is called *continuously differentiable*, if is
differentiable
and its
derivative
is

A function is called -times *continuously differentiable*, if it is -times differentiable, and its -th derivative is continuous.