(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 (3)f(x)??? 求f ?(0)和f ?(2) ? 5?x5 解 (1)y??cos x?sin x?
y?x????3?1?3?1? ?co?s?sin662226??2?2?2? ?co?s?sin44224 y? (2)
x??d??sin???cos??1sin??1sin???cos?? d?22d?d????
???co??1sins?1?2???2?2(1??)? 24442242424 (3)f?(x)?3?2x? ?3? f?(2)?17? f(0)?2515(5?x)25 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 ? (6)y??[(a211?122?1222?x)]?(a?x)?(a2?x2)?
22222?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?
2221?(1?2x)1?(1?2x)x?x2?1?1?11?222)]??(1?x)?(1?x2)?
(2)y??[(1?x2?x ??1(1?x2)2?(?2x)??
222(1?x)1?x?x?x?x22y??(e)?cos3x?e(cos3x)??e2(?3 (3)
x)?cos3x?e?2(?sin3x)(3x)?
2xx???sx?3e2sin3x??1e2(co3sx?6sin3x)? ??1e2co322xx|x|1? (1)???(?12)?22xxxx?11?(1)21?(1)2xx?1(1?lnx)?(1?lnx)12x?? (5)y??x?
(1?lnx)2x(1?lnx)2 (4)y???1 (6)y??cos2x?2?x2?sin2x?1?2xcos2x2?sin2x?
xx (7)y??111?(x)???1?? 222x2x?x21?(x)1?(x) (8)y??111?(x?a2?x2)???[1?(a2?x2)?] x?a2?x2x?a2?x22a2?x2111? ?[1?(2x)]?222222x?a?x2a?xa?x ?21secxtanx?secx?secx? (9) y???(secx?tanx)??secx?tanxsecx?tanx21?cscxcotx?cscx?cscx? ? (10) y???(cscx?cotx)?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)] ?
同济大学第六版高等数学上册课后答案全集
