微积分全英微积分全英 (35).pdf

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1、12.2 Partial DerivativesRecall knownDefinition of Derivative0The derivative of a function at is the following limit:fx0000()()()limxf xxf xfxx+=Derivative:the rates of change Recall:Derivative of =()in 3 stepsStep3Step2Step10limxzzx =Find limit:z()()f xxf xxx+=Find ratio:z()()f xxf x=+Find increment

2、:Derivative to partial derivative=,function of two variables,=,0function of one single variable ,0=0For(,):partial derivative with respect to at(0,0)For():derivative at 0=0if is fixed as and has the increment ,then has (,)zf x y=xx0yyDefinitionDefinition of partial derivative the corresponding parti

3、al incrementIf the limit0000(,)(,)=+xzf xx yf x yexists,is defined in some neighborhood of the point (,)zf x y=00(,),xy000000(,)(,)limlim +=xxxzf xx yf xyxxDefinitionDefinition of partial derivative Denote itor 00000000|,|,|or(,)x xx xxx xxy yy yy yzfzfx yxx=00(,).xyat the pointxthen it is called th

4、e partial derivative of with respect to(,)zf x y=DefinitionDefinition or,xzfzxx(,)xfx yIf has the partial derivative with respect toat any point(,)zf x y=xNote:The result is still a function of two variables and .xyof the domain,Dthen denote the partial derivative with respect to xDefinitionDefiniti

5、on 000000(,)(,)limlim +=yyyzf xyyf xyyyor,00yyxxyz=,00yyxxyf=,00yyxxyz=).,(00yxfyDenote itat is given 00(,)xySimilarly,the partial derivative of with respect to (,)zf x y=yDefinitionor,yzfzyy(,).yfx y(,)zf x y=with respect to .The notations are Similarly,define the partial derivative of the function

6、yDefinition Note:,is a function of two variables,Calculation MethodIdea:Such as,findonly need put as a constant,y(,)xfx ythen make use of the derivative method of the function of one variable with changeablexNote:Partial derivative of a multivariate function does not need a new calculation method.Ex

7、ample 1:Calculation questionFind the partial derivative of about x.32yfxex y=+(,)xfx y=ye223x y+Example 2If,find and .(0)yzx x=zxzy,1=yyxxzxxyzyln=Tips:When finding the partial derivative of one variable,treat the other variables as constants Example 3If ,prove .pVRT=1pVTVTp=Proof：;2VRTVp=;pRTV=;RVp

8、T=RTpV=RTVp=pVTR=pTTVVp2VRT pR RV.1=pVRT=Note:The sign of the partial derivative just whole mark not the quotient of the numerator and the denominatorExample 4(,).f x yIf 22(,)(0,0),(,)0(,)(0,0).xyx yxyf x yx y+=find partial derivatives of(1)When(,)(0,0),x y y222)(yx+)(22yx+xxy 2 ,)()(22222yxxyy+=(,

9、)xfx y=2222222222()2().()(,)yxxyxyyx xyxyxyfx y+=+=+Example 4(,)f x y.(,).f x yIf 22(,)(0,0),(,)0(,)(0,0).xyx yxyf x yx y+=find partial derivatives of(2)When (,)(0,0),x y=by the definiton,we get(,)(0,0)x y=But not continuous at .Note:By the above calculation,has partial derivatives at(,)f x y(,)(0,0

10、);x y=00lim0=xx0(0,0)(0,0)limxfxfx+=(0,0)xf=)0,0(yf00lim0=yy=+yfyfy)0,0()0,0(lim00(,)(,)(,)limyyf x yy zf x y zfx y zy+=Expand to 3 or more variables:0(,)(,)(,)limxxf xx y zf x y zfx y zx+=0(,)(,)(,)limzzf x y zzf x y zfx y zz+=The definition of partial derivatives can be expanded to the functions o

11、f 3 or more variables.Such as,at (,)uf x y z=(,)x y zExample 5Find second the four partial derivatives ofIf find and .(),23,f x y zxyyzzx=+,xyffzfSimilarly,Thus,To get ,think of and as constants,xfwith respect to the variabley,xzand differentiate(),3xfx y zyz=+(),2yfx y zxz=+(),23.zfx y zyx=+Example

12、 6If ,find three partial derivatives at 2(,)()sinlnxyf x y zzax=()1,0,2.Method 1:)2,0,1(zf002=z=)2,0,1(xf)2,0,(xf12lnsin=xx12lncos2=xxx2=)2,0,1(yf000=yMethod 2:1,0,2=,(1,0,2)Definition of Higher Partial DerivativesThe four second order partial derivatives of are as follows:(),zf x y=xzx 22xz =),(yxf

13、xx=),(yxfxy=2zy x=xzy ),(yxfyy=22yz =yzy ),(yxfyx=2zx y=yzx ,:,(,)functions of variables and What are the partial derivatives of,(,)?Definition of Higher Partial DerivativesOne of the third partial derivatives of (),zf x y=Definition The partial derivatives of second order and above are called highe

14、r-orderpartial derivatives.=Question:How many third order partial derivatives for function(,)?Example 7Find the four second partial derivatives of()32,sinyxf x yxex yy=+()221,cos3yxxfx yex yyy=+()32,cos2yyxxfx yxex yyy=+()221,sin6xxxfx yxyyy=+()23432,sincos2yyyxxxxfx yxexyyyy=+Example 7Find the four

15、 second partial derivatives of()32,sinyxf x yxex yy=+()221,cos3yxxfx yex yyy=+()32,cos2yyxxfx yxex yyy=+=+=+，：mixed partial derivativesExample 822ln,=+zxyIf prove it satisfy the equation:22220.zzxy+=Proof:22220.zzxy+=So),ln(2122yx+22lnzxy=+=,22yxxxz+=22222222 222 2()2,()()zxyxxyxxxyxy+=+By the symme

16、try of and xy,)(2222222yxyxyz+=Geometric meaning of partial derivative00Let(,),M xy(,).N x y0000()()tanlim()xxf xf xfxxx=The slope of MT:x0 x TxyO)(xfy=CN M,:the slope of(,)zf x y=0 xyTxT0yyzO),(0yxfz=P),(0yxfz=x,=,=,:the slope of SummaryDefinition：DerivativeDefinition：Partial derivativeDefinition：Higher-order partial derivatives Geometric meaning of partial derivative for bivariate functionPartial Derivatives