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:

$\frac{d}{dx}[f(x)+g(x){]}_{x=x_0}=f^{\prime }(x_0)+g^{\prime }(x_0)$

or equivalently:

$(f+g{)}^{\prime }(x_0)=f^{\prime }(x_0)+g^{\prime }(x_0)$

In point-free notation:

$(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$

• 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:

$\frac{d}{dx}[\lambda f(x)]{|}_{x=x_0}=\lambda f^{\prime }(x_0)$

In terms of generalized linearity

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Related rules

• 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