Let $D$ be a subset of the real numbers $\mathbb R$ and let $f:D\to\mathbb R$ be a function which is differentiable on $D.$ If its derivative $f':D\to\mathbb R$ is itself differentiable at a point $x\in D,$ then its derivative $$\frac{d^2f(x)}{dx}:=f'\;'(x):=(f')'(x)$$ is called the second-order derivative of $f$ at $x.$
In general, if $f:D\;\cap\;]x-\epsilon,x + \epsilon[\to\mathbb R$ is $(k-1)$ times differentiable in $D\;\cap\;]x-\epsilon,x + \epsilon[$, and if its $(k-1)$-th derivative is differentiable at $x$, then we call $f^{(k)}$ its $k$-th derivative (or the derivative of order $k$ at $x.$)^{1}
There are several equivalent notations of this derivative:
$$f^{(k)}(x):=\frac{d^kf(x)}{dx^k}:=\left(\frac d{dx}\right)^kf(x):=\frac d{dx}\left(\frac{d^{k-1}f(x)}{dx^{k-1}}\right).$$
Definitions: 1
Proofs: 2
Propositions: 3
The derivative of order $0$ of $f$ is the function $f$ itself. ↩