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: $\! \frac{d^ky}{dx^k}$

and is defined as: $\! \frac{d^ky}{dx^k} = \frac{d}{dx}\left[\frac{d}{dx} \left[ \dots \left[\frac{d}{dx}(y)\right] \dots \right] \right]$

where the $d/dx$ occurs $k$ times. Alternatively we can define it inductively as: $\! \frac{d^ky}{dx^k} = \frac{d}{dx} \frac{d^{k-1}y}{dx^{k-1}}$

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