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:
with the base case .
Suppose , so is a dependent variable depending on , the independent variable. The derivative of with respect to is denoted:
and is defined as:
where the occurs times. Alternatively we can define it inductively as:
with the base case being defined as .