函數(shù)y(74x²+120x+111)=4的多階導數(shù)計算過程

y=eq\f(4,74x2+120x+111)的一階和二階導數(shù)主要內(nèi)容：本文通過函數(shù)商的求導和函數(shù)乘積的求導法則，介紹求函數(shù)y=eq\f(4,74x2+120x+111)一階及二階導數(shù)的主要過程和步驟。一、兩種方法求一階導數(shù)函數(shù)商求導法：y=eq\f(4,74x2+120x+111)y′=eq\f(4′(74x2+120x+111)-4(74x2+120x+111)′,(74x2+120x+111)2),=eq\f(0-592x-480,(74x2+120x+111)2),=-16*eq\f(37x+30,(74x2+120x+111)2).函數(shù)乘積求導法：y=eq\f(4,74x2+120x+111)，即：y(74x2+120x+111)=a,兩邊同時對x求導得：y′(74x2+120x+111)+y(148x+120)=0,y′(74x2+120x+111)=-y(148x+120)y′=-yeq\f(148x+120,74x2+120x+111),則：y′=-16*eq\f(37x+30,(74x2+120x+111)2).函數(shù)導數(shù)定義法：y=eq\f(4,74x2+120x+111)，則：y′=lim(t→0)eq\f(eq\f(4,74(x+t)2+120(x+t)+111)-eq\f(4,74x2+120x+111),t),=lim(t→0)eq\f(74x2+120x+111-74(x+t)2-120(x+t)-111,(74x2+120x+111)[74(x+t)2+120(x+t)+111]t),=lim(t→0)eq\f(-74(2xt+t2)-120t,(74x2+120x+111)[74(x+t)2+120(x+t)+111]t)=lim(t→0)eq\f(-74t(2x+t)-120,(74x2+120x+111)[74(x+t)2+120(x+t)+111]t),=lim(t→0)eq\f(-74(2x+t)-120,(74x2+120x+111)[74(x+t)2+120(x+t)+111]),=eq\f(-74(2x+0)-120,(74x2+120x+111)[74(x+0)2+120(x+0)+111]),=-16*eq\f(37x+30,(74x2+120x+111)2)。二、函數(shù)商求二階導數(shù)直接求導法：∵y′=-16eq\f(37x+30,(74x2+120x+111)2)∴y′′=-16*eq\f(37(74x2+120x+111)-2(37x+30)(148x+120),(74x2+120x+111)3),=16*eq\f(2(37x+30)(148x+120)-37(74x2+120x+111),(74x2+120x+111)3),=48*eq\f(2738x2+4440x+1031,(74x2+120x+111)3)。函數(shù)乘積求導法：∵y′=-16eq\f(37x+30,(74x2+120x+111)2),即：y′(74x2+120x+111)2=-16(37x+30)∴y′′(74x2+120x+111)2+y′*4*(74x2+120x+111)*74x=-16*37，y′′(74x2+120x+111)2=-16*37-y′*4*(74x2+120x+111)*74x，將y′=-16eq\f(37x+30,(74x2+120x+111)2)代入上式，有：y′′(74x2+120x+111)2=-16*37+16eq\f(37x+30,(74x2+120x+111)2)*4*(74x2+120x+111)*74x，y′′(74x2+120x+111)2=-16*37+16eq\f(37x+30,74x2+120x+111)*4*74x，化簡得：y′′=48*eq\f(2738x2+4440x+1031,(74x2+120x+111)3)。導數(shù)定義法：∵y′=-16eq\f(37x+30,(74x2+120x+111)2)∴y′′=-16EQ\s(lim,t→0)eq\f(eq\f(37(x+t)+30,[74(x+t)2+120(x+t)+111]2)-eq\f(37x+30,(74x2+120x+111)2),t),=-16EQ\s(lim,t→0)eq\f([37(x+t)+30](74x2+120x+111)2-(37x+30)[74(x+t)2+120(x+t)+111]2,t(74x2+120x+111)2[74(x+t)2+120(x+t)+111]2),=16EQ\s(lim,t→0)eq\f((37x+30){[74(x+t)2+120(x+t)+111]2-(74x2+120x+111)2}-37t(74x2+120x+111)2,t(74x2+120x+111)2[74(x+t)2+120(x+t)+111]2),=16EQ\s(lim,t→0)eq\f((37x+30)t(74t+2*74x+120)[74(x+t)2+120(x+t)+111+74x2+120x+111]-37t(74x2+120x+111)2,t(74x2+120x+111)2[74(x+t)2+120(x+t)+111]2)=16EQ\s(lim,t→0)eq\f((37x+30)(74t+2*74x+120)[74(x+t)2+120(x+t)+111+74x2+120x+111]-37(74x2+120x+111)2,(74x2+120x+111)2[74(x+t)2+120(x+t)+111]2)=16*eq\f(2*(37x+30)(2*74x+120)(74x2+120x+111)-37(74x2+120x+111)2,(74x2+120x+111)2(74x2+120x+111)2)