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 $f^{'''\dots '}$ where ${}^{'}$ 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	$\!f'$	$\!f^{(1)}$	$\!f'$	the derivative, also called the first derivative
2	$\!f''$	$\!f^{(2)}$	$\!(f')^{'}$	the second derivative
3	$\!f'''$	$\!f^{(3)}$	$\!((f')^{'})^{'}$	the third derivative

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

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

or as:

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

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:

$\!{\frac {d^{k}y}{dx^{k}}}$

and is defined as:

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

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

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

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