# Higher derivative

## Contents

## Definition

### Terminology

**Higher derivatives** are also called **repeated derivatives** or **iterated derivatives**.

### Function and prime notation

Suppose is a function and is a nonnegative integer. The derivative of , denoted or where occurs a total of times, is defined as the function obtained by differentiating a total of times (i.e., taking the derivative, then taking the derivative of that, and so on, a total of times). The first few cases are shown explicitly:

Value of | Notation with repeated primes for | notation | Definition | In words |
---|---|---|---|---|

0 | the original function | |||

1 | the derivative, also called the first derivative
| |||

2 | the second derivative | |||

3 | the third derivative |

We could also define the derivative inductively as:

or as:

with the base case .

### Leibniz notation

Suppose , so is a *dependent variable* depending on , the *independent variable*. The derivative of with respect to is denoted:

or as:

and is defined as:

where the occurs times. Alternatively we can define it inductively as:

with the base case being defined as .