函數(shù)y(120x²+20x+20)=44的多階導(dǎo)數(shù)計(jì)算過程

y=eq\f(44,51x2+120x+20)的一階和二階導(dǎo)數(shù)主要內(nèi)容：本文通過函數(shù)商的求導(dǎo)和函數(shù)乘積的求導(dǎo)法則，介紹求函數(shù)y=eq\f(44,51x2+120x+20)一階及二階導(dǎo)數(shù)的主要過程和步驟。一、兩種方法求一階導(dǎo)數(shù)函數(shù)商求導(dǎo)法：y=eq\f(44,51x2+120x+20)y′=eq\f(44′(51x2+120x+20)-44(51x2+120x+20)′,(51x2+120x+20)2),=eq\f(0-4488x-5280,(51x2+120x+20)2),=-264*eq\f(17x+20,(51x2+120x+20)2).函數(shù)乘積求導(dǎo)法：y=eq\f(44,51x2+120x+20)，即：y(51x2+120x+20)=a,兩邊同時(shí)對x求導(dǎo)得：y′(51x2+120x+20)+y(102x+120)=0,y′(51x2+120x+20)=-y(102x+120)y′=-yeq\f(102x+120,51x2+120x+20),則：y′=-264*eq\f(17x+20,(51x2+120x+20)2).函數(shù)導(dǎo)數(shù)定義法：y=eq\f(44,51x2+120x+20)，則：y′=lim(t→0)eq\f(eq\f(44,51(x+t)2+120(x+t)+20)-eq\f(44,51x2+120x+20),t),=lim(t→0)eq\f(51x2+120x+20-51(x+t)2-120(x+t)-20,(51x2+120x+20)[51(x+t)2+120(x+t)+20]t),=lim(t→0)eq\f(-51(2xt+t2)-120t,(51x2+120x+20)[51(x+t)2+120(x+t)+20]t)=lim(t→0)eq\f(-51t(2x+t)-120,(51x2+120x+20)[51(x+t)2+120(x+t)+20]t),=lim(t→0)eq\f(-51(2x+t)-120,(51x2+120x+20)[51(x+t)2+120(x+t)+20]),=eq\f(-51(2x+0)-120,(51x2+120x+20)[51(x+0)2+120(x+0)+20]),=-264*eq\f(17x+20,(51x2+120x+20)2)。二、函數(shù)商求二階導(dǎo)數(shù)直接求導(dǎo)法：∵y′=-264eq\f(17x+20,(51x2+120x+20)2)∴y′′=-264*eq\f(17(51x2+120x+20)-2(17x+20)(102x+120),(51x2+120x+20)3),=264*eq\f(2(17x+20)(102x+120)-17(51x2+120x+20),(51x2+120x+20)3),=264*eq\f(2601x2+6120x+4460,(51x2+120x+20)3)。函數(shù)乘積求導(dǎo)法：∵y′=-264eq\f(17x+20,(51x2+120x+20)2),即：y′(51x2+120x+20)2=-264(17x+20)∴y′′(51x2+120x+20)2+y′*4*(51x2+120x+20)*51x=-264*17，y′′(51x2+120x+20)2=-264*17-y′*4*(51x2+120x+20)*51x，將y′=-264eq\f(17x+20,(51x2+120x+20)2)代入上式，有：y′′(51x2+120x+20)2=-264*17+264eq\f(17x+20,(51x2+120x+20)2)*4*(51x2+120x+20)*51x，y′′(51x2+120x+20)2=-264*17+264eq\f(17x+20,51x2+120x+20)*4*51x，化簡得：y′′=264*eq\f(2601x2+6120x+4460,(51x2+120x+20)3)。導(dǎo)數(shù)定義法：∵y′=-264eq\f(17x+20,(51x2+120x+20)2)∴y′′=-264EQ\s(lim,t→0)eq\f(eq\f(17(x+t)+20,[51(x+t)2+120(x+t)+20]2)-eq\f(17x+20,(51x2+120x+20)2),t),=-264EQ\s(lim,t→0)eq\f([17(x+t)+20](51x2+120x+20)2-(17x+20)[51(x+t)2+120(x+t)+20]2,t(51x2+120x+20)2[51(x+t)2+120(x+t)+20]2),=264EQ\s(lim,t→0)eq\f((17x+20){[51(x+t)2+120(x+t)+20]2-(51x2+120x+20)2}-17t(51x2+120x+20)2,t(51x2+120x+20)2[51(x+t)2+120(x+t)+20]2),=264EQ\s(lim,t→0)eq\f((17x+20)t(51t+2*51x+120)[51(x+t)2+120(x+t)+20+51x2+120x+20]-17t(51x2+120x+20)2,t(51x2+120x+20)2[51(x+t)2+120(x+t)+20]2)=264EQ\s(lim,t→0)eq\f((17x+20)(51t+2*51x+120)[51(x+t)2+120(x+t)+20+51x2+120x+20]-17(51x2+120x+20)2,(51x2+120x+20)2[51(x+t)2+120(x+t)+20]2)=264*eq\f(2*(17x+20)(2*51x+120)(51x2+120x+20)-17(51x2+120x+20)2,(51x2+120x+20)2(51x2+120x+20)2)