Higher Order Derivatives

Definition 4.4.1  
The second derivative of a function $f : \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: x \mapsto f(x)$ is defined as

$$f^{(2)} = (f')'$$ (4.4.1)

As is to be expected, this definition is extended to any given order by

$$f^{(n)} = (f^{(n-1)})'$$ (4.4.2)

Higher order derivatives are written as

$$\frac{d^n f}{dx^n}(x) \hbox{\ \ or \ \ } f^{(n)}$$ (4.4.3)

Never write something like $f^2 (x)$, because this could just as well mean $f(x) \cdot f(x)$ as it could mean $f'' (x)$.

Remark 4.4.2  
The natural question to ask is if a differentiable function is twice differentiable and so on. This is not the case. Take e.g.

$$f : \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: \... ...{if $x \leq 0$} \\ x \mapsto \frac{1}{2} x^2 & \text{if $x \geq 0$} \end{cases}$$ (4.4.4)

Note that the derivative of $f$ is the absolute value function, which itself is not differentiable. So $f$ is once differentiable, but not twice differentiable.
On the other hand, there are functions that are infinite order differentiable, like e.g. $x^3$ (or any polynomial for that matter).

Higher Order Derivatives

Function	$n$-th Derivative
$x^m$	$\frac{m!}{(m-n)!} x^{m-n}$
$\ln(x)$	$(-1)^{n-1} (n-1)! \frac{1}{x^n}$
$e^{kx}$	$k^n e^{kx}$
$\sin(x)$	$\sin\left(x+\frac{n\pi}{2}\right)$
$\cos(x)$	$\cos\left(x+\frac{n\pi}{2}\right)$
$\sinh(x)$	$\sinh(x)$ for $n$ odd
	$\cosh(x)$ for $n$ even
$\cosh(x)$	$\cosh(x)$ for $n$ odd
	$\sinh(x)$ for $n$ even

The $n$-th derivative of $\sin$ and $\cos$ functions might appear a bit strange at first glance, but this is just a ``shorthand" way of writing

$$\begin{aligned}\frac{d{}}{dx} \cos(x) &= -\sin(x) = \cos\left(x... ...{dx^4} \cos(x) &= \phantom{-}\cos(x) = \cos(x+2\pi) \end{aligned}$$