Order (derivative)

The order of a derivative is the number of successive differentiations performed to obtain the derivative from the original function. Formally, using Lagrange's notation for the derivative, the order-$k$ derivative is defined recursively by \begin{align*} f^{(0)}(x) &= f(x), \\ f^{(k)}(x) &= (f^{(k-1)}(x))'. \end{align*}

The function $f(x)$ itself is the order-$0$ derivative, the ordinary (first) derivative has order $1$, the second derivative has order $2$, and so on. In general, the derivative of order $k$ is called the $k$th derivative.