查看“︁微分的線性”︁的源代码

{{Copy edit|time=2020-08-22T19:15:41+00:00}}
{{微积分学}}
在微积分中，函数的任何线性组合的导数等于函数的导数的相同线性组合<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|accessdate=2020-08-22|archive-date=2021-04-29|archive-url=https://web.archive.org/web/20210429150734/https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA177|dead-url=no}}.</ref>，此属性称为'''微分的线性（linearity of differentiation）'''<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|accessdate=2020-08-22|archive-date=2021-04-29|archive-url=https://web.archive.org/web/20210429150755/https://books.google.com/books?id=OisInC1zvEMC&pg=PA71|dead-url=no}}.</ref>、线性法则（rule of linearity）、或微分的[[叠加原理|叠加法则]]<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|accessdate=2020-08-22|archive-date=2021-04-29|archive-url=https://web.archive.org/web/20210429150735/https://books.google.com/books?id=C8DiBQAAQBAJ&pg=PA89|dead-url=no}}</ref>。导数的基本属性是将两个简单的微分法则封装在一起：求和法则（两个函数之和的导数是导数的和）和常数法则（函數的常數倍的導數是該函數的導數的常數倍）<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|accessdate=2020-08-22|archive-date=2021-04-29|archive-url=https://web.archive.org/web/20210429150804/https://books.google.com/books?id=trC-jTRffesC&pg=PA259|dead-url=no}}.</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|accessdate=2020-08-22|archive-date=2021-04-29|archive-url=https://web.archive.org/web/20210429150749/https://books.google.com/books?id=1WLNBQAAQBAJ&pg=PA184|dead-url=no}}.</ref>。因此，可以说微分作用是线性的，或者[[微分算子]]是线性的算子<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|accessdate=2020-08-22|archive-date=2021-04-29|archive-url=https://web.archive.org/web/20210429150750/https://books.google.com/books?id=xP0RFUHWQI0C&pg=PA103|dead-url=no}}.</ref>。

== 說明 ==
設 {{math|''f''}} 和 {{math|''g''}} 為函數，同時 {{math|''α''}} 和 {{math|''β''}} 是常數，思考：

: <math>\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) + \beta \cdot g(x) ) </math>

通過微分的求和法則：

: <math>\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) ) + \frac{\mbox{d}}{\mbox{d} x} (\beta \cdot g(x))</math>

通過微分的常數法則，這一式子變為：

: <math>\alpha \cdot f'(x) + \beta \cdot g'(x)</math>

進而：

: <math>\frac{\mbox{d}}{\mbox{d} x}(\alpha \cdot f(x) + \beta \cdot g(x)) = \alpha \cdot f'(x) + \beta \cdot g'(x)</math>

忽略括號，這常被寫作

: <math>(\alpha \cdot f + \beta \cdot g)' = \alpha \cdot f'+ \beta \cdot g'</math>


== 參考資料 ==
{{Reflist|2}}

[[Category:包含证明的条目]]
[[Category:微分学]]
[[Category:求导法则]]
[[Category:分析定理]]
[[Category:微積分定理]]