Higher derivative
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 ![]() |
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Definition | In words |
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0 | ![]() |
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the original function |
1 | ![]() |
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the derivative, also called the first derivative |
2 | ![]() |
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the second derivative |
3 | ![]() |
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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
.