Higher derivative

Definition

Terminology

Higher derivatives are also called repeated derivatives or iterated derivatives.

Function and prime notation

Suppose $f$ is a function and $k$ is a nonnegative integer. The $k^{th}$ derivative of $f$, denoted $f^{(k)}$ or $f^{{\text{'}}^{″}{\dots }^{\prime }}$ where ${\text{'}}^{}$ occurs a total of $k$ times, is defined as the function obtained by differentiating $f$ a total of $k$ times (i.e., taking the derivative, then taking the derivative of that, and so on, a total of $k$ times). The first few cases are shown explicitly:

Value of $k$	Notation with repeated primes for $f^{(k)}$	$f^{(k)}$ notation	Definition	In words
0	$f$	$f^{(0)}$	$f$	the original function
1	$f^{\prime }$	$f^{(1)}$	$f^{\prime }$	the derivative, also called the first derivative
2	$f^{″}$	$f^{(2)}$	$(f^{\prime }{)}^{\text{'}}$	the second derivative
3	$f^{‴}$	$f^{(3)}$	$((f^{\prime }{)}^{\text{'}}{)}^{\text{'}}$	the third derivative

We could also define the $k^{th}$ derivative inductively as:

$f^{(k)}=(f^{(k-1)}{)}^{\text{'}}$

or as:

$f^{(k)}=(f^{\prime }{)}^{(k-1)}$

with the base case $f^{(0)}=f$.

Leibniz notation

Suppose $y=f(x)$, so $y$ is a dependent variable depending on $x$, the independent variable. The $k^{th}$ derivative of $y$ with respect to $x$ is denoted:

$\frac{d^k}{(dx{)}^k}y$

or as:

$\frac{d^{k}y}{d{x}^k}$

and is defined as:

$\frac{d^{k}y}{d{x}^k}=\frac{d}{dx}[\frac{d}{dx}[\dots [\frac{d}{dx}(y)]\dots ]]$

where the $d/dx$ occurs $k$ times. Alternatively we can define it inductively as:

$\frac{d^{k}y}{d{x}^k}=\frac{d}{dx}\frac{d^{k-1}y}{d{x}^{k-1}}$

with the base case $k=0$ being defined as $\frac{d^{0}y}{d{x}^0}=y$.