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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This article gives a statement of the form that a certain operator from a space of functions to another space of functions is a linear operator, i.e., applying the operator to the sum of two functions gives the sum of the applications to each function, and applying it to a scalar multiple of a function gives the same scalar multiple of its application to the function.
In terms of additivity and pulling out scalars
The following are true:
- 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:
- Repeated differentiation is linear
- Product rule for differentiation
- Product rule for higher derivatives
- Chain rule for differentiation
- Chain rule for higher derivatives
- Definite integration is linear
- Indefinite integration is linear
- Limit is linear: This states that the limit of the sum is the sum of the limits and scalars can be pulled out of limits.
We prove here the two-sided versions. Analogous proofs exist for the one-sided versions, and these use the one-sided versions of Fact (1).
Proof of additivity
Given: and are functions that are both differentiable at .
To prove: is differentiable at , and
Proof: Our proof strategy is to start out by trying to compute as a difference quotient, and keep simplifying this, using Fact (1) in the process.
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