Math 2.0

Real functions Theorem

Linearity of differentiation

Builds on DerivativeLimit of a sum of sequences

Statement

Let ff and gg be real functions on an open interval II, both differentiable at a∈Ia \in I. Let α\alpha and β\beta be real numbers. Then αf+βg\alpha f + \beta g is differentiable at aa, and (αf+βg)′(a)=αf′(a)+βg′(a).(\alpha f + \beta g)'(a) = \alpha f'(a) + \beta g'(a).

Remarks

The difference quotient of αf+βg\alpha f + \beta g is α\alpha times that of ff plus β\beta times that of gg. The limit laws give the result. The case α=β=1\alpha = \beta = 1 is the sum rule (f+g)′=f′+g′(f + g)' = f' + g'. The case β=0\beta = 0 is the constant multiple rule. Mathlib states the sum rule as deriv_add.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionDerivative, definition, no evidence levelDerivativedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementLimit of a sum of sequences, statement, no evidence levelLimit of a sum of seq…statementLinearity of differentiation, statement, no evidence levelLinearity of differen…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Derivative definition
  6. Limit of a sequence definition
  7. Epsilon-N definition of the limit of a sequence statement
  8. Limit of a sum of sequences statement
  9. Linearity of differentiation statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Linearity_of_differentiation, Statement and derivation
About this page
Tag0M0TKFRX2A
URIm2://0M0TKFRX2A
Aliasessum-rule-for-derivatives
Kindstatement, theorem
WritingW0 stub
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0M0TKFRX2A.
UsesDerivative, Limit of a sum of sequences
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