Let the real function $f$ be defined and differentiable on the open interval $I$ . Then for every $x \in I$ , there exists the value $f'(x)$ as a certain real number. This means that we have a new function

(1)

Forming the derivative function of a function is called differentiation, the corresponding verb is differentiate.

If the derivative function $f'$ is differentiable on $I$ , then we have again a new function, the derivative function of the derivative function of $f$ , which is denoted by $f''$ . Then $f$ is said to be twice differentiable. Formally, $$f''(x) = \lim_{h\to 0}\frac{f'(x+h)-f'(x)}{h}\quad \mathrm{for\,all\,}\,x\in I.$$ The function $x\mapsto f''(x)$ is called the second order derivative or the second derivative of $f$ . Similarly, one can call (1) the first (order) derivative of $f$ .

Example. The first derivative of $x\mapsto x^3$ is $x\mapsto 3x^2$ and the second derivative is $x\mapsto 6x$ , since $$\frac{d}{dx}(3x^2) = 2\cdot 3x^{2-1} = 6x.$$

If also $f''$ is a differentiable function, its derivative function is denoted by $f'''$ and called the third (order) derivative of $f$ , and so on.

Generally, $f$ can have the derivatives of first, second, third, ..., $n$ th order, where $n$ may be an arbitrarily big positive integer. If $n$ is four or greater, the $n$ th derivative of $f$ is usually denoted by $f^{(n)}$ . In addition, it's sometimes convenient to think that the $0$ th order derivative $f^{(0)}$ of $f$ is the function $f$ itself.

The phrase ``$f$ is infinitely differentiable'' means that $f$ has the derivatives of all orders.