Repeated differentiation is linear
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.
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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:
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: