函数f(x)=sinx/x如何用严格的定义求导函数(不是直接套公式)?

设y为x的增量,z为f的增量,则z=sin(x+y)/（x+y）-sinx/x=（sinxcosy+cosxsiny）/（x+y）-sinx/x=[x（sinxcosy+cosxsiny）-sinx]/[x(x+y)]
z/y=[xsinxcosy+xcosxsiny-xsinx-ysinx)]/[xy(x+y)]=[xsinx(cosy-1)+xcosxsiny-ysinx)]/[xy(x+y)]=sinx(cosy-1)/[y(x+y)]+cosxsiny/[y(x+y)]-sinx/[x(x+y)]
当y趋于0时,几个等价无穷小量之间是可以替代的,比如siny等价于y,（cosy-1）等价于-y^2/2,用这几个等价无穷小量替代,可以得到原式等于
-ysinx/(x+y)+cosx/(x+y)-sinx/[x(x+y)]=0+cosx/x-sinx/x^2=(xcosx-sinx)/x^2
完毕
关键是用到等价无穷小量,这些等价式在极限里用处很大,要牢记