![高等应用数学](https://wfqqreader-1252317822.image.myqcloud.com/cover/247/26179247/b_26179247.jpg)
2.6.1 微分的概念
定义 设函数y=f(x)在点x0处可导,任给自变量x在x0处有改变量Δx,当Δx有微小改变量时,把f'(x0)Δx称为函数y=f(x)在点x0处的微分,记作,即
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069008.jpg?sign=1738871715-KqeOJ5pQpDZ7iI04uJUkQysjHiU499DK-0-9649e49d9c377c47cd1a35d22792b745)
此时称函数y=f(x)在点x0处可微.
例1 如图2-4所示,一块正方形金属薄片受温度变化影响,其边长由x0变化到x0+Δx时,
(1)求此薄片的面积在边长x0处的微分;
(2)求此薄片的面积的改变量;
(3)求此薄片的面积在边长x0处的微分与改变量相差多少.
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069001.jpg?sign=1738871715-uUQehKzAoF9PbnKZTj3sQtCClmCYvYgf-0-a9d1e4624743528a611880d19a447a9d)
图 2-4
解 此薄片的面积函数为S=x2.
(1)由微分的定义,得在边长x0处的微分
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069009.jpg?sign=1738871715-ydFZrdJDhuyytQmCDCq9TpehIvccuXQB-0-be19a15fb54a5e0c634603b4a78a11b6)
(2)边长由x0变化到x0+Δx时,此薄片的面积的改变量为
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069002.jpg?sign=1738871715-HzUENfZsTABFe0k3D3v7mnmBUR5Pbvtg-0-79490191e128780a5f510c3c85274f3c)
(3)薄片的面积在边长x0处的微分与改变量相差
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069010.jpg?sign=1738871715-pZjz8W0OcizkkRryuOxwy2yNKMd0iZ4U-0-b11c857f6b58048c386f30438b31efb6)
在例1中,如果x0=3,Δx=0.01,ΔS=0.0601,,它们相差0.0001.
一般地,随着Δx的绝对值越来越小,即当Δx→0时,Δy与dy之间是什么关系?它们相差多少?对此有下面的定理:
定理1 若函数y=f(x)在点x0处可微,则当f'(x0)≠0,且Δx→0时,Δy与dy是等价无穷小,即Δy~dy.
证明 因为函数y=f(x)在点x0处可微,则
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069011.jpg?sign=1738871715-9oWM0vSTIBUPnO3gHf6iR65nqCjewLGp-0-7432761ad0afc0acd44b4fcbc991cbca)
且函数y=f(x)在点x0处连续、可导,于是Δx→0时,Δy→0, ,即它们都是无穷小.
又因为f'(x0)≠0,所以
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00069003.jpg?sign=1738871715-gq9MzGLfpVDlwtVlZlaKkhwTsUY307So-0-3f33b157afd56f53f7d66eae3803ae7b)
则Δx→0时,Δy与dy是等价无穷小,即Δy~dy.
定理2 若函数y=f(x)在点x0处可微,则当Δx→0时,Δy-dy=ο(Δx).
证明 因为函数在点x0处可微,所以函数y=f(x)在点x0处可导,则
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00070001.jpg?sign=1738871715-yGPw9hCCtUEp1spuBfFIscAstJG14IzJ-0-4bb1ff5d4b9a3e899ac034e7a3e52b51)
根据具有极限函数与无穷小的关系,推得
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00070002.jpg?sign=1738871715-BUNh09j0lzou8Do5ScoxsMMdc7CNJxxc-0-3be5b1aa776cd2e1ae09cc0894a31e7d)
Δy=f'(x0)Δx+α(Δx)Δx.
移项,得Δy-f'(x0)Δx=α(Δx)Δx,
且 α(Δx)Δx=ο(Δx).
将 代入上式,得
Δy-dy=ο(Δx).
发现:(1)因为当Δx→0时,Δy与dy是等价无穷小且Δy-dy=ο(Δx),所以Δy≈dy.
(2)当y=x时,由函数微分定义,得dy=dx=(x)'·Δx=1·Δx=Δx,则称自变量x的改变量Δx称为自变量的微分,记作dx,于是
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00070007.jpg?sign=1738871715-tVa1brAnhNHwXCNz1Eihd8NFvWDh7HQi-0-22c9b8a26be846fe99b86c1578f7e8f1)
若函数y=f(x)在某区间内每一点都可微,则称函数y=f(x)在此区间内可微,且dy=f'(x)dx.因为dx≠0,因此,所以,函数的导数是函数的微分与自变量微分的商,简称微商.