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

is called the -th 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

continuous.

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