Repeated differentiation is linear
From Calculus
This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules
Contents
Statement
For a positive integer, denote by
the function obtained by differentiating
a total of
times. The operation
is a linear operator. We give two equivalent ways of stating this below.
In terms of additivity and pulling out scalars
The following are true:
- Repeated differentiation is additive, or
derivative of sum is sum of derivatives: If
and
are functions that are both differentiable at
, we have:
or equivalently:
In point-free notation:
- Constants (also called scalars) can be pulled out of differentiations: If
is differentiable at
and
is a real number, then:
In terms of generalized linearity
Suppose are functions that are all differentiable at a point
and
are real numbers. Then: