1、Section 9.3Fermat2Partial Derivatives of a Function of Two VariablesThe horizontal coordinate in this plane is x;the vertical coordinate is z.We define the partial derivativeof f with respect to x at the point00(,)xyas the ordinary derivativewith respect to x at0(,)f x yofthe point 0.xx the vertical
2、00(,)xyIf(,),f x yis a point in the domain of a function will cut the surface 0yy(,)zf x y 0(,).zf x y in the curve plane3Partial Derivatives of a Function of Two VariablesDefinition(Partial Derivative偏导数偏导数 with Respect to x)00(,)U xyWe fix the independent variable y at y0,i.e,00(,).xyof0.yy and,x
3、When the independent variable x has an increment0000(,)(,)xx yU xy the corresponding function f(x)has an0000(,)(,).f xx yf xy If the limit00000(,)(,)limxf xx yf xyx exists,then this limit is called the partial derivative of the functionis defined in a neighbourhoodSuppose that the function(,)zf x y(
4、,)zf x y with respect to x at the point 00(,).xyincrement4Partial Derivatives of a Function of Two VariablesDefinition(continued)Partial Derivative with Respect to xis denoted byi.e.The partial derivative of functionrespect to x at(,)zf x y 00(,)xy0000000(,)(,)(,)lim.xxf xx yf xyfxyx 00000000(,)(,)(
5、,),(,),or xxxyxyzffxyzxyxx“(similar to the lowercase Greek letter“”used in the The stylized“limit definition)is just another kind of“d”.It is convenient to have thisdistinguishable way of extending the Leibniz differential notation intoa multivariable context.5Partial Derivatives of a Function of Tw
6、o Variables00(,)xy0000(,)(,),yxyzzxyy 0000000(,)(,)(,)lim.yyf xyyf xyfxyy Similarly,we can define the partial derivative of the function with respect to y as followsat the point or(,)zf x y 00(,).xyfy It may also be denote by6Partial Derivatives of a Function of Two Variables(,)zf x y to x at every
7、pointIf the partial derivative of the function derivative 2(,)Rx yD(,)xfx ycalled the partial derived(,),x yis also a function of two variablefunction偏导函数 with respect to x,denoted by,.or xxzffzxxwith respectexists,then the partial7Partial Derivatives of a Function of Two VariablesExample Finding Pa
8、rtial Derivatives at a Pointwe treat x as a constant and differentiate with respect to y:with respect to x:(4,5)/fy3(4)113.is The value of at 2(31)fxxyyyy03110 x 31.x2(,)31.f x yxxyyifFind the values of andat the point/fx(4,5)/fySolution/fxTo find,we treat y as a constant and differentiate 2(31)fxxy
9、yxx23 100 xy 23.xy/,fySimilarly,to find(4,5)/fx2(4)3(5)is The value of at 7.8Partial Derivatives of a Function of Two VariablesSolution We treat x as a constant and f as a product of y and sin:xy(sin)fyxyxy sinyxyy (sin)xy(cos)()yxyxyy (sin)()xyyy cossin.xyxyxy/,fx(,)sin.f x yyxy DIYwhere Find theEx
10、ample Finding Partial Derivatives as a FunctionifFind/fy(,)sin.f x yyxy Partial Derivatives of a Function of Two Variables9Example Suppose that Prove that.yzx 12.lnyzzzx xx yProof 1,ln.yyzzyxyxxxx Substituting them into the left of given equation,we have 111ln22.lnlnyyyyzzyyxxxxzx xx yxx This is the
11、 conclusion.Finish.10Partial Derivatives of a Function of Two Variables Example Discuss the existence of partial derivatives of the function222222,0,(,)0,0,xyxyxyf x yxy SolutionSince(0,0)(0,0)0,fxfx (0,0)(0,0)0.fyfy The partial derivatives of f at the originboth exist and(0,0)0,(0,0)0.xyffat the or
12、igin.11Partial Derivatives of a Function of Two VariablesDefinition If both partial derivatives of the function at(,)f x yis partial00(,).xythe point(,)f x yexist,then we say that the function derivable at the point 00(,)xyNote is derivable at()f xWe know that if a single variable functionbut this m
13、ay 0,xx 0 xx implies that the function f is continuous at As an example,discuss the not be true with functions of two variables.222222,0,(,)0,0 xyxyxyf x yxy at pointcontinuity of function(0,0).Total Differentials 全微分全微分Definition(Total differential)Suppose that a function(,)zf x y If for00(,)Uxyis
14、defined in a neighbourhoodof the point 00(,).xycan be expressed in the form0000(,)(,)zf xx yyf xy and22,yxy where 12,a aare constants independent of x is an infinitesimal of higher order with respect to as()o and0,0000(,)(,),xx yyUxy the increment of the function f at 00(,)xy12(),zaxayo 12axay is ca
15、lled the total differential of the function f at the pointandthen the function f is said to be differentiable at the point 00(,),x y00(,),xyor00(,).df xydenoted by 00(,)xydz12Total Differentials 全微分全微分Obviously,when r is sufficiently small,the total differential is the linearand main part of the inc
16、rement of the function f at the point00(,).xyDefinition(continued)(Total differential)ThusWe define the differentials of the independent variables to be equal to0012(,).xydza dxa dythen the total differential of 00(,)xy,;xdxydy their increments,that iscan be written asthe function f at the point 001
17、2(,).xydzaxay 1.What conditions will differentiable function satisfy?and2?a 2.If a function is differentiable,what are the values of1a13Total DifferentialsTheorem(Necessary Conditions for Differentiability)Then 00(,);xy(1)f must be continuous at the point is differentiable at a point Suppose that a
18、function(,)zf x y 00(,).xyare expressed2a100200(,),(,),xyafxyafxywhere1aandexist and000000(,)(,)(,).xyxydzfxy dxfxy dy(2)both partial derivatives of the function f at the point 00(,)xy00(,)xyisthe point that is,the total differential of the function f atby 0012(,),xydza dxa dy1415Total Differentials
19、Proof of(1)f must be continuous at the point then the expressionWe have12()zaxayo holds.i.e.0,(0,0).xy Let0lim0,z or00000,0lim(,)(,).xyf xx yyf xy 00(,).xyis continuous at the point(,)f x yThus,If the function f is differentiable at the point00(,),xy16Total DifferentialsProof of(2)The value of a1 an
20、d a2 are just the partial derivatives of f 12().zaxayo 00001(,)(,)(),f xx yf xyaxox so thatthen(,)zf x y Since 00(,),xyis differentiable at the pointwe have0,y Let Notice that 1ais independent of.x,we have0 x Letting 00001(,)(,)().f xx yf xyoxaxx 00000(,)(,)limxf xx yf xyx 1.a 00(,)xfxy Similarly,we
21、 have002(,).yfxya 10()limxoxax 17Total Differentials In this case,the total differential of the function f at the point(,)x y,anddz can be denoted by dfor.xydzf dxf dyDefinition(Differentiable Function)is differentiable at every point in the region(,)zf x y If the function2R,is the then f is said to
22、 be a differentiable function in.If domain of the function f,then f is called a differentiable function.18Total DifferentialsExample Discuss the differentiability of the function 222222,0,(,)0,0,xyxyf x yxyxy at the point(0,0).Solution By the definition of the partial differential of the function,th
23、en(0,0)(0,0)0,xyffit is easy to see that(0,0)(0,0)xyzfxfy 22.()()xyxy 19Total DifferentialsSolution(continued)Thus 220lim()()xyxy It is easy to see that the limit22()()xyxy 22.()()xyxy This means that the given function is not differentiable does not exist.at the point(0,0).(0,0)(0,0)xyzfxfy 20Total
24、 DifferentialsProof 0000(,)(,)zf xx yyf xy 0000(,)(,),f xyyf xy Since 0000(,)(,)f xx yyf xyy Theorem(Sufficient Condition for Differentiability)both exist(,),zf x y zy If the partial derivatives of a function zx and00(,).xydifferentiable at the point then the function f isin the neighbourhood of poi
25、nt 00(,)xyand are continuous at the point00(,)xy21Total DifferentialsBy the Lagrange theorem we have and because the partial derivatives is continuous at point 00(,),xythen0000(,)(,)f xx yyf xyy 010(,),f xx yyxx 010(,)f xx yyxx 001(,),xfxyxx Similarly,0000(,)(,)f xyyf xy 002(,)f xyyyy 002(,),yfxyyy
26、as0.y where 20 as0.x where 10 Proof(continued)000000000000(,)(,)(,)(,)(,)(,)zf xx yyf xyf xx yyf xyyf xyyf xy 22Total Differentials000012(,)(,).xyzfxyxfxyyxy Thus,Proof(continued)Since 12xy 1212xy0,0,xy as0Hence 00000(,)(,)lim0.xyzfxyxfxyy Therefore,f is differentiable at the point00(,).xy0000001000
27、0002(,)(,)(,),(,)(,)(,).xyf xx yyf xyyfxyxxf xyyf xyfxyyy 23Total DifferentialsExample Find the total differentials of function xyze at the pointSolution Since,xyzyex ,xyzxey 2(2,1),zex 2(2,1)2,zey then(2,1)Therefore,the total differentials of the given function at point is22(2,1)2.dze dxe dy000000(
28、,)(,)(,)xyxydzfxy dxfxy dy(2,1).24Higher-Order Partial Derivatives高阶偏导数高阶偏导数(x),xR.nufSuppose that w.r.t.ifx If the partial derivative of derivative(or second partial derivative二阶偏导数)of the function f at the pointexists,thenjxat0 xw.r.t.0 xdenoted by ixfirst and then,jx020 x x(x),jijiffxxxx 0(x),(1,
29、1).ijfinjnor0(x),ijx xfthis partial derivative is called the second order partial25Higher-Order Partial Derivativeshas the four second(,)zf x y For example,a function of two variablesorder partial derivatives222222,.xxxyyxyyffffffxxxyxy xffffffxyx yyyy Pure second partial derivatives 二阶纯二阶纯偏导数偏导数Mix
30、ed second partial derivatives 二阶混合二阶混合偏导数偏导数26The Applications of the Total Differential to Approximate Computation and Estimation of ErrorsIf a function f of n variables is differentiable at point x0,then we have000(xx)(x)(x)(),fffdfo 001(x)(x)(x).inxiifffx or00001(xx)(x)(x)(x),inxiiffdffx we haves
31、o if|x|1,The right side of the expression is a linear function of the n variables.this is called a hyperplane.xR,3,nnIf 2xR,this is a plane;if 1,the increment of function f This expression shows that when can be approximated by its total differential.27The Applications of the Total Differential to A
32、pproximate Computation and Estimation of Errors1)Approximate computation of functional valuesExample Find the approximate value of 33(1.97)(1.01).Solution Let 3300(,),(,)(2,1),0.03,0.01.f x yxyxyxy Then(2,1)yf233(2,1)32yxy 1.2(2,1)xf233(2,1)32xxy 3,Thus,0000(,)(,)f xydf xy33(1.97)(1.01)2.945.00(,)f
33、xx yy (2,1)(2,1)(2,1)xyffxfy 28The Applications of the Total Differential to Approximate Computation and Estimation of Errors2)Estimation of errorsLet the measured value of x and y be x0 and y0,andquantities x and y.the maximum absolute errors of the measurement are x and y.Thus,|,|.xyxy000(,),zf xy
34、 which is obtained by Then the value through measurement of theIf z is determined by a function(,)zf x y(,)zf x y using the approximate valuecomputation from the formula is also an approximate value for the quantity z.0,y0 xandThe question is what the error is,when we replace the actual value zwith
35、the approximate value z0.00|(,)(,)|?zf x yf xy29The Applications of the Total Differential to Approximate Computation and Estimation of Errors are very small,the approximate equality|y Since|x andzdz can be used.000000000(,)(,).|(,)(,)yzxxyfxyfxyzf xyf xy|zdz0000|(,)(,)|xyfxyxfxyy 0000|(,)|(,)|xyfxy
36、xfxyy0000|(,)|(,)|,xxyyfxyfxyso that the absolute error of the value may be taken as0z0zmay be taken asand the relative error of 0000|(,)|(,)|,zxxyyfxyfxy30andproduced by the computation of z from the values approximate value The Applications of the Total Differential to Approximate Computation and
37、Estimation of ErrorsFind the absolute error and relative error of the.zxy Example Let 0.y0zof the measurement 0 xSolution substituting these into the expression,xyzy zxSince of the absolute error and the relative error,we obtain 0000|(,)|(,)|,zxxyyfxyfxythe relative error:000.|yzxzxy the absolute er
38、ror:00|,zxyyx000000000(,)(,)|(,)(,)yzxxyfxyfxyzf xyf xy(the relative error w.r.t.x and y)31Higher-Order Partial DerivativesSimilarly,the nth order partial derivatives of a function f may be definedFor examplefrom the partial derivatives of n1-st order of the function f.2.ijkx x xkjiffxxx The second
39、order or higher order partial derivatives are called by the joint name higher-order partial derivative,and the partial derivatives of the function are sometimes called first-order partial derivatives of thefunction or partial derivatives for short.The operations of calculating higher-order partial d
40、erivatives are actually the operations of calculating derivatives of functions of one variables.32Higher-Order Partial DerivativesExample Find all second derivatives of the function(0).yzxxThe first order derivatives are Solution 1,ln.yyzzyxxxxy Another derivation with respect to x and y respectivel
41、y gives the secondpartial derivatives2zzx yxy (1)(1)ln,yyyxxx22zzyyy 2(ln).yxx 2zzy xyx (1)(1)ln,yyxyxx22zzxxx 2(1),yy yx 22zzy xx y 33Higher-Order Partial DerivativesExample Let22222222,0,(,)0,0.xyxyxyf x yxyxy Prove that(0,0)(0,0).xyyxff Proof It is easy to find the first order partial derivatives
42、22222222222224,0,(,)()0,0.yxyx yxxyfx yxyxyxy 22222222222224,0,(,)()0,0;xxyx yyxyfx yxyxyxy 34Higher-Order Partial DerivativesProof(continued)Thus Again,from the definition of the partial derivative we have(0,),(,0).xyfyyfxx 1.(0,0)yxf0(,0)(0,0)limyyxfxfx 0limxxx 1,(0,0)xyf0(0,)(0,0)limxxyfyfy 0limy
43、yy Thus(0,0)(0,0).xyyxff 35Higher-Order Partial DerivativesIn general,the order of derivation may affect the result,but it is possible to prove that if fxy and fyx are both continuous at a point P then fxy=fyx at a point P.If all the mth This result can be extended to higher partial derivatives.order mixed partial derivatives are continuous,then the order of derivatives are continuous,then the order of derivation does not affect For instance,if all third order mixed partial derivatives of the result.are continuous,then we have(,)f x y z the function.xyzyzxzxyyxzxzyzyxffffff;xxyxyxyxxfff
