1、Rules for Finding DerivativesDerivation rules for sum,difference,product and quotient of functions2(Derivation rules of rational operations)Suppose that the function f,:Rg I xI are derivable at;then their sum,difference,product and quotientare all derivable at x,and()()()()fgxfxg x(1)()()()()()()fgx
2、fx g xf x g x(2)2()()()()()()0)()ffx g xf x g xxg xggx (3)In particular()()()cfxcfx Rc (is a constant),21()()()0)()fxxf xffx .Derivation rules for sum,difference,product and quotient of functions3Lets prove only the rule(2).()()()()yf xx g xxf x g x By the definition of the derivative,we have()()yf
3、x g x Let ,then Proof:()()()()()()()()f xx g xxf x g xxf x g xxf x g x 0()()limxyfgxx Since is derivable at ,()g xxit must be continuous at x and henceThereforeFinish.()()()()()()fgxfx g xf x g x.0lim()()xg xxg x .00lim()lim()xxfgg xxf xxx .()()g xxff xg .Derivation rules for sum,difference,product
4、and quotient of functions4Find the derivative of223cosln.xyxxxxSolution:By rules(1)and(2),we have Finish.2()(2)(3cos)(ln)xdyxxxxdx22 ln23sin()ln(ln)xxxxxxxln22 ln23sin2xxxxxxxln122 ln23sin2xxxxxx.Derivation rules for sum,difference,product and quotient of functions2(tan)secxx 2(cot)cscxx 5tanyx coty
5、x Find the derivative of and .Solution:By the quotient rule,we have2(sin)cos(cos)sincosxxxxx 2(tan)secxx 2(cot)cscxx That is By the same way,we haveFinish.2222cossinseccosxxxx sin(tan)cosxxx Derivation rules for sum,difference,product and quotient of functions62xyx Find the equation for the tangent
6、to the curve at the point(1,3).Solution The slope of the curve is 222()1.yxxx Then the slope at x=1 is 1()1.xy The line through(1,3)with slope m=-1 is 3(1)yx Finish.or4.yxDerivation rules for sum,difference,product and quotient of functions00lim()lim()(0).xxf xf xf7Solution:Determine the constants a
7、 and b,such that the functionis continuous and derivable on 223,0,(),0,xxxf xaxbx (,).Notice that f(x)is a polynomial function while x 0 or x 0.1(ln|)(ln);xxxWhen x 0,then prove that ()()()()()ln()v xv xu xu xv xu x .Examples192(ln(12)dyxxxdx,dydxExample 1:Findwhere2ln(12).yxxxSolution:221(12)12xxxx
8、xx22122(1)122 2xxxxxx21.2xxFinish.Examples20()()ln()(),v xv xu xu xe()()ln()()v xv xu xu xe ()()()ln()v xu xv xu x ()ln()()ln()v xu xev xu x ()u x()v xExample 2:If and are both derivable on I,and u(x)0,then prove that()()()()()ln()v xv xu xu xv xu x .Proof:Since we have Finish.Examples21 ()()()()()l
9、n()v xv xu xu xv xu x lnxxxxxx 1(ln)xxxxx,dydx(0).xyxxExample 3:FindwhereSolution:(ln1).xxxFinish.xxx Could you find?Examples222.yaxbxcExample 4:If axis x is the tangent line of a parabola,What conditions will a,b and c satisfy?02bxa 20.22bbabcaaSolution:()yf x Since axis x is the tangent line of ,t
10、hen the tangent point0(,0)xwill have the formand it is satisfying thatandTherefore00 x xdyydx (2)020axb(3)By(3),we have0 xand substitute in(1),we have2000axbxc(1)Finish.23(1)()0c (where c is a constant)(2)()lnxxaaa (0a ,1a )(3)()xxee (4)1()()xxR (5)(sin)cosxx (6)(cos)sinxx (7)2(tan)secxx (8)2(cot)cscxx (9)(sec)sectanxxx (10)(csc)csc cotxxx (11)1(ln)xx (12)1(log)(0,1)lnaxaaxa (13)21(arcsin)1xx (14)21(arccos)1xx (15)21(arctan)1xx (16)21(arccot)1xx (17)(sinh)coshxx (18)21(arcsin)1hxx (19)(cosh)sinhxx (20)21(arccos)1hxx (21)2(tanh)1/coshxx (22)21(arctan)1hxx
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