講義成果講稿

第二 導(dǎo)數(shù)的運(yùn)算法習(xí) 2-(cotx)csc2x (cscx)cscxcotx解 (cotx)(cosx)(cosx)sinx(sinx)cosx csc2x(cscx)(

) sin2x

sin2(sinx)cscxcotx

sin2 y

7 yln2x2xx33

y2cscxcotx yx3log2x 2

yexarccosxylnxx1 yxlnxcosx y1x2 yxaax(a0); s1sint.1cos解 y(2x3)(3)(7)6x26

y(ln2x)(2x)(x)(2x)2xln2112xln2 y(2cscx)(cotx)2cscxcotxcsc2xy(ex)arccosxex(arccos )ex(arccosx )1exarccosx111y(x3)log 3x2log

x2(3logx ) x(logx

y(lnx)xlnx1lnx y(x2)lnxcosxx2(lnx)cosxx2lnx(cos

2xlnxcosxxcosxx2lnxsinxy(1x2)(1x2)(1x2)(1x2)(1x2

(1x2

y(xa)axxa(ax)axa1axxaaxlnaxa1ax(xlnaa)s(1sint)(1cost)(1sint)(1cost)1costsint 以初速v豎直上拋的物體,s與時間tsvt1gt2 求該物體的速度v(t);解(1)v(tsv0gt(2)物體到達(dá)最高點(diǎn)時,v(t)0,即

gt0,從而tv0g 解該點(diǎn)為(e,0),所求切線的斜率為 (lnx1

ln

1,切線方程為:yxe,法線方程為

yxeye3x2 ycos(43x2) yarctan(ex) y(arcsinx)2 yatanx2(a0) ycos2(tan3x)

sin2xy x y x解 ye3x(3x2)6xe3xysin(43x2)(43x2)6xsin(43x2)(ex y 1e2x 1e2x1y2(arcsinx)(arcsinx)21yatanxlna(tanx2)atanxlnasec2x2(x2)2lnaxsec2x2atany2cos(tan3x)[cos(tan3x)]2cos(tan3x)sin(tan3x)(tan33sin(2tan3x)tan2x(tanx)3tan2xsec2xsin(2tan3x)sin2 sin2 y xln ) x2 (sin sin2 1 ln2sin2 x2ln cos() x

x x1ln 1lnx y(e )e (lnx)xx lnx ) x xf()sin3 ,013y1arcsin2x,x 3 y3e5x5(1x),x1解 12 1解 f()3cos3

(12

3cos ,f 4(12

y1arcsin2x

,

16 π)x13xx13x y15e5x5,yx15(13e5)yln(secxtanx) ylntanx arctan(tan)1111 ysinnxcosnx y sin

y

y xx1 yxarcsinx2

ysh2ch3x44 ylnchx 2ch2

yaaxxaa

axa解

secxtanxsec2y secxsecxtansec2 1sec2y2 2 cscx tanx 11tan2x 13cos2x ynsinn1xcoxcosnxnsinnxsinnxnsinn1xcos(n1)x 1111(1x)2x(1y

2xcosx2sin2xsinx22sinxcosxsin4

2xcosx2sinx2sinx2cos.sin31111x2(x1x2(x1x2y (12(x1x2yarcsinxx 2

arcsinx21 4

24 y

x(

)ch3x

sh3x

chch3xx

sh3xxyshx2shxthx(11)th3x

ch2yaaxlna(ax)aaxaa1axalna(xaln2aaxaaaaxa1alnaxa1axf(xg(x都可導(dǎo),y的導(dǎo)數(shù)dy yf(ex)ef(x) yf(sin2x)f(cos2x) ylnf(x)arctang(x2) y f2(x) 解 yf(ex)exef(x)f(ex)ef(x)f(x)f(ex)ef(x)xf(ex)eff2(x) yf(sin2x)2sinxcosxf(cos2x)2sinxcossin2x[f(sin2x)f(cos2x)]f(x)f(x)2f(x)f(x)2y f(x) x1g2(x2 f(x 1g2(x2x2f(x)f2f(x)f(x)22f2(x) 4f(x)f(x)g(x)4f2(x) g(x)f(x在(ll內(nèi)可導(dǎo),證明:f(x是偶函數(shù),f(x是奇函數(shù);如果f(x)是奇函數(shù),則f(x)是偶函數(shù). 如果f(x)是偶函數(shù),則有f(x)f(x)