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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Statement

In terms of additivity and pulling out scalars

The following are true:

• Differentiation is additive, or derivative of sum is sum of derivatives: If $f$ and $g$ are functions that are both differentiable at $x=x_{0}$, we have:

${\frac {d}{dx}}[f(x)+g(x)]_{x=x_{0}}=f'(x_{0})+g'(x_{0})$

or equivalently:

$(f+g)'(x_{0})=f'(x_{0})+g'(x_{0})$

In point-free notation:

$(f+g)'=f'+g'$

• Constants (also called scalars) can be pulled out of differentiations: If $f$ is differentiable at $x=x_{0}$ and $\lambda$ is a real number, then:

${\frac {d}{dx}}[\lambda f(x)]|_{x=x_{0}}=\lambda f'(x_{0})$

In terms of generalized linearity

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Related rules

• 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