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Concept
Linearity of differentiation
Résumé
In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions; this property is known as linearity of differentiation, the rule of linearity, or the superposition rule for differentiation. It is a fundamental property of the derivative that encapsulates in a single rule two simpler rules of differentiation, the sum rule (the derivative of the sum of two functions is the sum of the derivatives) and the constant factor rule (the derivative of a constant multiple of a function is the same constant multiple of the derivative). Thus it can be said that differentiation is linear, or the differential operator is a linear operator.
Let f and g be functions, with α and β constants. Now consider
By the sum rule in differentiation, this is
and by the constant factor rule in differentiation, this reduces to
Therefore,
Omitting the brackets, this is often written as:
We can prove the entire linearity principle at once, or, we can prove the individual steps (of constant factor and adding) individually. Here, both will be shown.
Proving linearity directly also proves the constant factor rule, the sum rule, and the difference rule as special cases. The sum rule is obtained by setting both constant coefficients to . The difference rule is obtained by setting the first constant coefficient to and the second constant coefficient to . The constant factor rule is obtained by setting either the second constant coefficient or the second function to . (From a technical standpoint, the domain of the second function must also be considered - one way to avoid issues is setting the second function equal to the first function and the second constant coefficient equal to . One could also define both the second constant coefficient and the second function to be 0, where the domain of the second function is a superset of the first function, among other possibilities.)
On the contrary, if we first prove the constant factor rule and the sum rule, we can prove linearity and the difference rule.
Source officielle
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