2-2高等数学同济大学第六版本

习题 2?2 1? 推导余切函数及余割函数的导数公式? (cot x)???csc2x ? (csc x)???csc xcot x ? cosx?cosx 解 (cotx)??(cosx)???sinx?sinx?sinxsin2x22sinx?cosx??1??csc2x? ??22sinxsinxs??csc (csxc)??(1)???co2xx?coxt? sinxsinx 2? 求下列函数的导数? 7?2?12? (1)y?4?x5x4x (2) y?5x3?2x?3ex ? (3) y?2tan x?sec x?1? (4) y?sin x?cos x ? (5) y?x2ln x ? (6) y?3excos x ? (7)y?lnx? xxe (8)y?2?ln3? x (9) y?x2ln x cos x ? (10)s?1?sint? 1?cost7?2?12)??(4x?5?7x?4?2x?1?12)? 解 (1)y??(4?x5x4x28?2? ??20x?6?28x?5?2x?2??20?x6x5x2 (2) y??(5x3?2x?3ex)??15x2?2x ln2?3ex? (3) y??(2tan x ?sec x?1)??2sec2x?sec x?tan x?sec x(2sec x?tan x)? (4) y??(sin x?cos x)??(sin x)??cos x?sin x?(cos x)? ?cos x?cos x?sin x?(?sin x)?cos 2x? (5) y??(x2ln x)??2x?ln x?x2?1?x(2ln x?1) ? x (6) y??(3excos x)??3ex?cos x?3ex?(?sin x)?3ex(cos x?sin x)?

1?x?lnxx? ?1?ln (7)y??(lnx)??x2xxx2xx2xex(x?2)ee?x?e?2x? (8)y??(2?ln3)??? xx4x3 (9) y??(x2ln x cos x)??2x?ln x cos x?x2?1?cos x?x2 ln x?(?sin x) x 2x ln x cos x?x cos x?x2 ln x sin x ? (10)s??(1?sint)??1?cost

3? 求下列函数在给定点处的导数? (1) y?sin x?cos x ? 求y?x??6cost(1?cost)?(1?sint)(?sint)1?sint?cost?? (1?cost)2(1?cost)2和y?x??4? d? (2)???sin??1cos??求d?2???? 423x?? 求f ?(0)和f ?(2) ? (3)f(x)?5?x5 解 (1)y??cos x?sin x? y????3?1?3?1? ?cos?sinx??662226x?? y? (2)??2?2?2? ?co?s?sin44224d??sin???cos??1sin??1sin???cos?? d?22 d?d???????co??1sins?1?2???2?2(1??)? 24442242424 (3)f?(x)?3?2x? ?3? f?(2)?17? f(0)?(5?x)252515 4? 以初速v0竖直上抛的物体? 其上升高度s与时间t的关系是s?v0t?1gt2? 2求?

(1)该物体的速度v(t)? (2)该物体达到最高点的时刻? 解 (1)v(t)?s?(t)?v0?gt? (2)令v(t)?0? 即v0?gt?0? 得t?v0? 这就是物体达到最高点的时刻? g 5? 求曲线y?2sin x?x2上横坐标为x?0的点处的切线方程和法线方程? 解 因为y??2cos x?2x? y?|x?0?2? 又当x?0时? y?0? 所以所求的切线方程为 y?2x? 所求的法线方程为 y??1x? 即x?2y?0? 2 6? 求下列函数的导数? (1) y?(2x?5)4 (2) y?cos(4?3x)? (3)y?e?3x? (4) y?ln(1?x2)? (5) y?sin2x ? (6)y?a2?x2? (7) y?tan(x2)? (8) y?arctan(ex)? (9) y?(arcsin x)2? (10) y?lncos x?

解 (1) y??4(2x?5)4?1?(2x?5)??4(2x?5)3?2?8(2x?5)3? (2) y???sin(4?3x)?(4?3x)???sin(4?3x)?(?3)?3sin(4?3x)? (3)y??e?3x?(?3x2)??e?3x?(?6x)??6xe?3x? (4)y??12?(1?x2)??12?2x?2x2? 1?x1?x1?x (5) y??2sin x?(sin x)??2sin x?cos x?sin 2x ? 2222

(6)y??[(a211?122?1222?x)]?(a?x)?(a2?x2)? 2?1x22 ?(a?x)2?(?2x)??? 222a?x1 (7) y??sec2(x2)?(x2)??2xsec2(x2)? x1ex?(e)?? (8)y??? 1?(ex)21?e2x (9) y??2arcsinx?(arcsinx)??2arcsinx? 1?x2 (10)y??1?(cosx)??1(?sinx)??tanx? cosxcosx 7? 求下列函数的导数? (1) y?arcsin(1?2x)? (2)y?1? 1?x2 (3)?xy?e2cos3x? (4)y?arccos1? x (5)y?1?lnx? 1?lnx (6)y?sin2x? x (7)y?arcsinx? (8)y?ln(x?a2?x2)? (9) y?ln(sec x?tan x)? (10) y?ln(csc x?cot x)? 解 (1)y??1?2?(1?2x)????1? 1?(1?2x)21?(1?2x)2x?x2

(2)y??[(1?x2?1?1?11?222)]??(1?x)?(1?x2)? 2?1x2 ??(1?x)2?(?2x)?? 222(1?x)1?x?x?x?xy??(e2)?cos3x?e2(cos3x)??e2(?3 (3)x)?cos3x?e?2(?sin3x)(3x)? 2xx???1122sx?3esin3x??e2(co3sx?6sin3x)? ??eco322xx|x|1? (1)???(?1)?222xxxx?11?(1)21?(1)2xx?1(1?lnx)?(1?lnx)12x?? (5)y??x? 2(1?lnx)x(1?lnx)2 (4)y???1?sin2x? (6)y??cos2x?2?x2?sin2x?1?2xcos2xxx2 (7)y??111? ?(x)???1?222x2x?x21?(x)1?(x)111?(x?a2?x2)???[1?(a2?x2)?] x?a2?x2x?a2?x22a2?x2111?[1?(2x)]?? 222222x?a?x2a?xa?x (8)y?? ?21secxtanx?secx?secx?(secx?tanx)?? (9) y??? secx?tanxsecx?tanx21?cscxcotx?cscx?cscx???(cscx?cotx)? (10) y?? cscx?cotxcscx?cotx

8? 求下列函数的导数? (1)y?(arcsinx)2? 2 (2)y?lntanx? 2 (3)y?1?ln2x? (4)y?earctanx? (5)y?sinnxcos nx ? (6)y?arctanx?1? x?1 (7)y?arcsinx? arccosx (8) y=ln[ln(ln x)] ? (9)y1?x?1?x? 1?x?1?x (10)y?arcsin1?x? 1?x 解 (1)y??2(arcsinx)?(arcsinx)? 22 ?2(arcsxi)?n1?(x)? 221?(x)22 ?2(arcsxi)?n1?1? 221?(x)222arcsxin2 ? 24?x (2)y??1?(tanx)??1?sec2x?(x)? 2tanx22tanx22

cx?1?cscx? ?1?se2x22tan21?(1?ln2x)? (3)y??1?ln2x?21?ln2x11?2lnx?(lnx)???2lnx?1 ?x21?ln2x21?ln2xlnx ?? 2x1?lnx (4)y??earctanx?(arctanx)? ?earctan1x?11?(x)2?(x)? ?earctaxnarctaxn1e? ???21?(x)2x2x(1?x) (5) y??n sinn?1x?(sin x)??cos nx?sinnx?(?sin nx)?(nx)? ?n sinn?1x?cos x ?cos nx?sinnx?(?sin nx)?n ?n sinn?1x?(cos x?cos nx?sin x?sin nx)? n sinn?1xcos(n?1)x ? (6)y??(x?1)?(x?1)111? ?(x?1)?????22x?1x?1(x?1)1?x2x?121?()1?()x?1x?11arccosx?1arcsinx221?x1?x?y? (7) (arccosx)2 ? ? (8)y??1?arccoxs?arcsixn 2(arccox)s1?x2? 21?x(arccxo)s22?1?[ln(lnx)]??1?1?(lnx)? ln(lnx)ln(lnx)lnx1?1?1?1 ?? ln(lnx)lnxxxlnx?ln(lxn)1?1)(1?x?1?x)?(1?x?1?x)(1?1)21?x21?x (9)y??21?x21?x(1?x?1?x)2(

? (10)y??1? 221?x?1?x?(1?x)?(1?x)11 ?(1?x)???1?x(1?x)21?x1?x1?1?1?x1?x1? (1?x)2x(1?x) ?? 9. 设函数f(x)和g(x)可导? 且f2(x)?g2(x)?0? 试求函数y?f2(x)?g2(x)的导数? 解 y??1?[f2(x)?g2(x)]? 2f2(x)?g2(x) ?1?[2f(x)f?(x)?2g(x)g?(x)] 222f(x)?g(x) ?f(x)f?(x)?g(x)g?(x)? 22f(x)?g(x)dy? dx 10? 设f(x)可导? 求下列函数y的导数 (1) y?f(x2)? (2) y?f(sin2x)?f(cos2x)? 解 (1) y??f ?(x2)?(x2)?? f ?(x2)?2x?2x?f ?(x2)? (2) y??f ?(sin2x)?(sin2x)??f ?(cos2x)?(cos2x)? ? f ?(sin2x)?2sin x?cos x?f ?(cos2x)?2cosx?(?sin x) ?sin 2x[f ?(sin2x)? f ?(cos2x)]? 11? 求下列函数的导数? (1) y?ch(sh x )? (2) y?sh x?ech x? (3) y?th(ln x)? (4) y?sh3x ?ch2x ? (5) y?th(1?x2)?

(6) y?arch(x2?1)? (7) y?arch(e2x)? (8) y?arctan(th x)? 1? 2ch2x (10)y?ch2(x?1) x?1 解 (1) y??sh(sh x)?(sh x)??sh(sh x)?ch x ? (9)y?lnchx? (2) y??ch x?ech x?sh x?ech x?sh x?ech x(ch x?sh2x) ? 1?(lnx)?? (3)y??21? 2ch(lnx)x?ch(lnx) (4) y??3sh2x?ch x?2ch x?sh x ?sh x?ch x?(3sh x?2) ? (5)y??212?(1?x2)?2?2x2? ch(1?x)ch(1?x) (6)y??12x? ?(x2?1)??2421?(x?1)x?2x?22x12e2x?? ?(e)?2x24x(e)?1e?1 (7)y?? (8)y??11?1?1?1 ??(th x)?1?(thx)21?th2xch2x1?sh2xch2xch2x1 ?212?? 2chx?shx1?2shx (9)y??1?(ch x)??14?(ch2x)? ch x2chx ?sh x?14?2ch x?shx ch x2chx2sh xshxsh x?chx?shx?3? ? 3ch xchxchxsh x?(ch2x?1)sh3x?3?th3x? ?3chxchx (10)y??2ch(x?1)?[ch(x?1)]??2ch(x?1)?sh(x?1)?(x?1)? x?1x?1x?1x?1x?1

?sh(2?x?1)?x?1(x?1)?(x?1)2sh(2?x?1)? ?x?1(x?1)2(x?1)2 12? 求下列函数的导数? (1) y?e?x(x2?2x?3)? (2) y?sin2x?sin(x2)? (3)y?(arctanx)2? 2 (4)y?lnnx? xt?t (5)y?et?e?t? e?e (6)y?lncos1? x (7)?sin21xy?e? (8)y?x?x? (9) y?xarcsinx?4?x2? 2 (10)y?arcsin2t2? 1?t 解 (1) y???e?x(x2?2x?3)?e?x(2x?2) ?e?x(?x2?4x?5)? (2) y??2sin x?cos x?sin(x2)?sin2x?cos(x2)?2x ?sin2x?sin(x2)?2x?sin2x?cos(x2)? (3)y??2arctanx?12?1?24arctanx? 22x?421?x41?xn?lnx?nxn?11?nlnx? ? (4)y??xx2nxn?1(et?e?t)(et?e?t)?(et?e?t)(et?e?t)4e2t? (5)y???(et?e?t)2(e2t?1)21tan1? (6)y??sec1?(cos1)??sec1?(?sin1)?(?1)?xxxxxx2x2

(7)?sin21x?(?sin2y??e1)??e?sin2x?(?2sin1)?cos1?(?1) xxxx211?si2n12x? ?2?sin?exx (8)y??11?(x?x)???(1?1) 2x2x?x2x?x ?2x?1? 4x?x?x (9)y??arcsinx?x?21?1??(?2x)?arcsinx? 22224?x21?x412?(1?t2)?2t?(2t)12t1?(2)??? (10)y?? 221?t(1?t)1?(2t2)21?(2t2)21?t1?t22(1?t2)2(1?t2)1?t ?? ??22(1?t2)2|1?t2|(1?t2)(1?t)

(7)?sin21x?(?sin2y??e1)??e?sin2x?(?2sin1)?cos1?(?1) xxxx211?si2n12x? ?2?sin?exx (8)y??11?(x?x)???(1?1) 2x2x?x2x?x ?2x?1? 4x?x?x (9)y??arcsinx?x?21?1??(?2x)?arcsinx? 22224?x21?x412?(1?t2)?2t?(2t)12t1?(2)??? (10)y?? 221?t(1?t)1?(2t2)21?(2t2)21?t1?t22(1?t2)2(1?t2)1?t ?? ??22(1?t2)2|1?t2|(1?t2)(1?t)

本文来源：https://www.bwwdw.com/article/xs0w.html