Editing Linearity of differentiation (section)

{{Short description|Calculus property}}
In [[calculus]], the [[derivative]] of any [[linear combination]] of [[function (mathematics)|function]]s equals the same linear combination of the derivatives of the functions;<ref>{{citation|title=Calculus: Single Variable, Volume 1|first1=Brian E.|last1=Blank|first2=Steven George|last2=Krantz|publisher=Springer|year=2006|isbn=9781931914598|page=177|url=https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA177}}.</ref> this property is known as '''linearity of differentiation''', the '''rule of linearity''',<ref>{{citation|title=Calculus, Volume 1|first=Gilbert|last=Strang|publisher=SIAM|year=1991|isbn=9780961408824|pages=71–72|url=https://books.google.com/books?id=OisInC1zvEMC&pg=PA71}}.</ref> or the [[Superposition principle|superposition rule]] for differentiation.<ref>{{citation|title=Calculus Using Mathematica|first=K. D.|last=Stroyan|publisher=Academic Press|year=2014|isbn=9781483267975|page=89|url=https://books.google.com/books?id=C8DiBQAAQBAJ&pg=PA89}}.</ref> It is a fundamental property of the derivative that encapsulates in a single rule two simpler rules of differentiation, the [[sum rule in differentiation|sum rule]] (the derivative of the sum of two functions is the sum of the derivatives) and the [[constant factor rule in differentiation|constant factor rule]] (the derivative of a constant multiple of a function is the same constant multiple of the derivative).<ref>{{citation|title=Practical Analysis in One Variable|series=[[Undergraduate Texts in Mathematics]]|first=Donald|last=Estep|publisher=Springer|year=2002|isbn=9780387954844|pages=259–260|url=https://books.google.com/books?id=trC-jTRffesC&pg=PA259|contribution=20.1 Linear Combinations of Functions}}.</ref><ref>{{citation|title=Understanding Real Analysis|first=Paul|last=Zorn|publisher=CRC Press|year=2010|isbn=9781439894323|page=184|url=https://books.google.com/books?id=1WLNBQAAQBAJ&pg=PA184}}.</ref> Thus it can be said that differentiation is [[linear map|linear]], or the [[differential operator]] is a [[linear map|linear]] operator.<ref>{{citation|title=Finite-Dimensional Linear Algebra|series=Discrete Mathematics and Its Applications|first=Mark S.|last=Gockenbach|publisher=CRC Press|year=2011|isbn=9781439815649|page=103|url=https://books.google.com/books?id=xP0RFUHWQI0C&pg=PA103}}.</ref>