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.
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 .