1、Differential of a Function12Concept of Differential00()()yf xxf x we want to calculate the value of.y In many problems,we need to discuss the relation between()yf x of some functionx and of a dependent variable x,andExample 1where are both constant(,),yaxba b For linear function00()().ya xxbaxba x i
2、s a linear function with respect to y.x Concept of Differential300lim()xyfxx 0()()yfxxx 0()()yfxxxx 0()fxx ()yf x 0 xWe had known that if functionis derivable at ,we haveand thus0 x 0lim()0 xx where and .We multiply both sides of this equalityx by ,0()()fxxox ()()oxxx x 0 x where is an infinitesimal
3、 of higher order than as .y This implies that we can divide the expression ofinto two parts,and we cally as the linear and main part of()ox and as the infinitesimalx of higher order than .4Concept of DifferentialDefinition(Differential 微 分微 分)Suppose that the function 0:()fU xR.If there is a linear
4、function()Lxa x (aR is a constant independent of x),such that 00()()()f xxf xa xox then f is said to be differentiable可微的可微的 at 0 x,and a x is called the differential of f at 0 x.We denote a x as 0()df x or 0 x xdya x .If f is differentiable at every point on the interval I,then f is said to be diff
5、erentiable on I.5Concept of DifferentialTheorem The necessary and sufficient condition for the function 0:()fU xR to be differentiable at 0 x is that f is derivable at 0 x.In this case,00()()df xfx dx .6The geometric meaning of the differentialIt is well known that()yf x is a curve and the derivativ
6、e 0()fx represents the slope of the tangent line to the curve at the point 00(,()P xf x,i.e.tan.Therefore,0()tandyfx dxPNNT .()yf x 0 xPQdyy()ox）xyo 0 xx Tx N()yf x Therefore,function can be approximated by a linear function 000()()()yf xfxxx 0()U x in some.that is,the differential of the functiony=
7、f(x)at x0 is just the increment of thepoint P.000,i.e.()()()().ydyf xf xfxxx ordinate of the tangent to the curve at theIf|is small,x we have7Rules of operations on differentialsSince we have ,it is easy to prove the following formulae,that is()dyfx dx 1.The differentials of elementary functions 122
8、2()0()(sin)cos(cos)sin(tan)sec(cot)csc(sec)sectan(csc)csccot()ln()11(log)(ln)ln11(arcsin)(arccos)11xxxxad Cd xxdxdxxdxdxxdxdxxdxdxxdxdxxxdxdxxxdxd aaadxd ee dxdxdxdxdxxaxdxdxdxxx 22211(arctan)(cot)11dxdxdxdxdxxx 8Rules of operations on differentialsIf()yf u,where u is an independent variable,by the
9、definition of differential,we have ()dyfu du If u is also a differentiable function()ug x of another variable x,then by the chain rule,the differential of the composite function ()yf g x is ()()dyfu g x dx Because()g x dxdu ,we also have()dyfu du and this property is called the invariance of the dif
10、ferential form.2.The differentials of rational operations 2()()()d uvdudvd CuCduuvduudvd uvvduudvdvv 3.The differentials of composite function 9Rules of operations on differentials Find the differential of the functionsin(21).yxFinish.cos(21)2xdxSolution:21uxLet ,then we havecosdyudu 2cos(21)xdxFinish.221(1)1xxdydee 2ln(1).xye Find the differential of the functionBy the invariance of the differential from,we haveSolution:2221xxexdxe 2221()1xxe d xe 2221xxxedxe
