1、Higherorder DerivativesHigher-order derivatives2()()v ts tvelocity():v tdisplacement():s tacceleration():a t()()()a tv ts t()()a ts tsecond order derivativeHigher-order derivatives3()()()fxfx :RfI (Higher order derivatives)Suppose that the functionis derivable.:RfI If its derivativexI is also deriva
2、ble at,then f isf called twice derivable at x and the derivate of at x is called the secondderivative of f at x,and denoted by.If f is everywheretwice derivable on I,then f is said to be twice derivable on I andf iscalled second derivative or second derived function of f on I.In general,ifxI(1):RnfI
3、 is derivable at ,1n the derivative of order,then f iscalled n times derivable at x(1)nf and the derivative ofat x is called thederivative of order n of f at x,and is denoted by()(1)()()nnffx .Higher-order derivatives 4we can define derivability of order n on I and the derivative ofSimilarly,order n
4、 on I for the function f.(Higher order derivatives)The derivative of order n of the function()yf x is denoted by()nynnd ydxor .Nn ()nfIf is continuous on I,then f is said to be continuously derivable oforder n on I,()nCor a function of class()()nfCI on I,denoted by.()nCIf f is a function of classon
5、I for anyon I,denoted by,then f is said to be()C infinitely derivable on I,or a function of class()()fCI .Higher-order derivatives5Note If the n th order derivative of a function()f x in(on)an interval I(open or closed)is continuous,then we say that()()()nf xCI,where note()()nCI denote all functions
6、 with n th order continuous derivative.For example,the polynomial of degree n,1010()(0)nnnnP xa xa xaa is one element of()(,)nC .(Higher order derivatives)The second derivative and derivatives of order higher that second order are all called higher order derivatives高阶导数.Commonly,f is said to be the
7、first order derivative of f,and f itself is said to be the zero order derivative of f.Higher-order derivatives6 If y=sin2x,find and y.y Solution4sin2;yx 2cos2;yx8cos2.yx Finish.Higher-order derivatives7 Prove the following formulae for derivatives of order n:()()xnxee(1)()(sin)sin2nxxn(2)()(cos)cos2
8、nxxn(3)()()(1)(1)(R,0)nnxnxx (4)()1(1)!ln(1)(1)(1)(1)nnnnxxx (5)Higher-order derivatives8()(sin)sin2nxxn By means of mathematical induction we know that the formula(2)holds.(sin)cossin222xxx (1)(sin)sin2kxxk Proof We will prove the formulae(2)and(5).(2)Because(sin)cossin2xxx ()(sin)sin2kxxk Assume t
9、hatholds,thensin(1)2xk cos2xk Higher-order derivatives9()1(1)!ln(1)(1)(1)(1)nnnnxxx The other solution is left to you.(3)22311 2ln(1)(1)(1)1xxx ()1(1)!ln(1)(1)(1)(1)nnnnxxx Proof(continued)Similarly to(2)by means of mathematical induction,we can obtains1ln(1)1xx 211ln(1)11xxx (5)BecauseHigher-order
10、derivatives10()()()(),Rnnnuvuv ()()()0()(1)()()1()nnn kkknknnn knkknnuvC uvuvC uvC uvuv Suppose that the functions u and v are both derivableuv of order n.Then and uv are also derivable of order n,and moreover we have:(1)The linear property:(2)Leibniz formula:Higher-order derivatives11()2()2()()2(1,
11、2,100),kxkkxuxeek22(),xf xx e(100)().fx Let find()()2,()2,()0(3,4,100).kv xx vxvxk(100)(100)1(99)(100)100()()()()()()()fxux v xCux v xu x vx 10022992982100992100 22222!xxxexexe100222(1002475).xexxSolution22(),(),xu xev xxWe takeIt is easy to seeBy means of the Leibniz formula we obtain:Finish.Higher
12、-order derivatives12Finish.(1)(1)1(2)()()()(1)()()!()()nnnnnfxxaxn nxaxnxax(1)(1)()()()()limnnnxafxfafaxa ()()()nf xxax Suppose that ,If it does not exist,explain;if it exists,find it.(1)()()nxCI where.Does()()nfaexist?Proof(1)()()nxCI Since ,(1)()nfx exists.By Leibniz formula,we have(1)()0nfa It is follows that(1)()nfa.Then,by the definition of,we have!()na So,()()!()nfana .Higher-order derivatives132132yxx 2211(2)(1)yxx 2233(1)1 2(1)1 2(2)(1)yxx Let()().nyx,find the nth derivativeSolution1121yxxSince ,soand By the mathematic induction,we have()1111()(1)!(2)(1)nnnnyxnxx .
