Sum rule

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The sum rule is one of the basic rules of differential calculus in mathematics . It says that the sum of two differentiable functions can be differentiated again and that such a sum of functions can be differentiated by term.

rule

Let the functions and be defined in a common interval that contains the position . At this point , both functions can be differentiated. Then the function is also with

differentiable at the point , and it applies

.

example

The functions

are differentiable with the derivative functions

.

Hence the function

on differentiable with the derivative function

.

Inferences

  • Difference rule : If one considers the difference for functions and , which are differentiable in, it follows from the sum rule and the factor rule that in is differentiable and applies to the derivative .
  • Together with the factor rule, the following results: If in are differentiable functions and real constants, then the linear combination is in turn in differentiable with a derivative function that is differentiated in terms of terms
    .
  • It follows from this: The differentiable functions (on a given interval) form a real vector space , and the differentiation is a linear mapping from this vector space into the vector space of all functions.

literature

  • Harro Heuser: Textbook of Analysis Part 1 . 17th edition. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0777-9 .

Web links