习题2?4
1? 求由下列方程所确定的隐函数y的导数 (1) y2?2x y?9?0?? (2) x3?y3?3axy?0? (3) xy?ex?y ?? (4) y?1?xey?
解 (1)方程两边求导数得 2y y??2y?2x y? ?0 ? ?于是 (y?x)y??y? ? y??y? y?xdy? dx (2)方程两边求导数得
3x2?3y2y??2ay?3axy??0? ?于是 (y2?ax)y??ay?x2 ? ?
ay?x2 y??2?
y?ax (3)方程两边求导数得 y ?xy??e x?y(1?y?)? ?于是 (x?ex?y)y??ex?y?y? ?
ex?y?y y???
x?ex?y (4)方程两边求导数得 y???e y?xeyy?? 于是 (1?xe y)y???e y?
ey?y?? ?
1?xey在点(2a, 2a)处的切线方程和法线方程?
44 解 方程两边求导数得 2?
?? 2x3?2y3y??0?
33112求曲线x32?y32?a3于是 y???
?1x3?1y3
?
在点(2a, 2a)处y???1?
44所求切线方程为
y?2a??(x?2a)? 即x?y?2a?
442所求法线方程为
y?2a?(x?2a)? 即x?y?0?
44d2y 3? 求由下列方程所确定的隐函数y的二阶导数2?
dx22
(1) x?y?1??
(2) b2x2?a2y2?a2b2? (3) y?tan(x?y)? (4) y?1?xey?
解 (1)方程两边求导数得 2x?2yy??0?? y??x?
yy?xxy?xy?yy2?x2x1???? y???()??? yy2y2y3y3 (2)方程两边求导数得
2b2x?2a2yy??0? ?
2b y???2?x? ay2bx)y?x(??2y2y?xy?2abb y????2?2??2?
ayay22a2y2?b2x24bb ??2???23? aa2y3ay (3)方程两边求导数得
y??sec2(x?y)?(1?y?)?
se2c(x?y)1 y?? ?221?sec(x?y)cos(x?y)?12sin(x?y)?co2s(x?y)1? ???1?2?sin(x?y)y22(1?y2)221 y???3y??3(?1?2)??? yyyy5 (4)方程两边求导数得
y??e y?xe yy?? ?
yyyeee?? y??? 1?xey1?(y?1)2?yeyy?(2?y)?ey(?y?)ey(3?y)y?e2y(3?y) y???? ??223(2?y)(2?y)(2?y) 4? 用对数求导法求下列函数的导数?
(1) y?(x)x?
1?x
(2)y?55x?5?
x2?2x?2(3?x)4 (3)y??
(x?1)5 (4)y?xsinx1?ex?? 解 (1)两边取对数得
ln y?xln|x|?xln|1?x|， 两边求导得
11(?x)?x?1? y??lnx?x??ln1yx1?x于是 y??(x)x[lnx?1]?
1?x1?x1?x (2)两边取对数得
lny?1ln|x?5|?1lnx(2?2)?
525两边求导得
111?1?2x y???? ?
y5x?525x2?2于是 y??1555x?5?[1?1?2x]?
2x2?2x?55x?2 (3)两边取对数得 lny?1lnx(?2)?4ln3(?x)?5lnx(?1)?
2两边求导得
1y??1?4?5?
y2(x?2)3?xx?1x?2(3?x)41?4?5] 于是 y??[2(x?2)x?3x?1(x?1)5 (4)两边取对数得
lny?1lnx?1lnsinx?1ln1(?ex)?
224两边求导得
x111et? y???cox? y2x24(1?ex)xx1?ex[1?1coxt?ex] 于是 y??xsin2x24(1?e)x1ex2xsinx1?e[?2cotx?x]? ?4xe?1 5? 求下列参数方程所确定的函数的导数
dy? dx?x?at2 (1) ?? 2y?bt??x??(1?sin?) (2) ??
y??cos??2dyy?解 (1)?t?3bt?3bt?
dxxt?2at2adyy?(2)???cos???sin??
?1?sin???cos?dxx??x?etsint,?时dy的值? 6? 已知?求当t?t3dx?y?ecost.dyyt?etcost?etsintcost?sint??? 解 ? dxxt?etsint?etcostsint?cost1?3dy221?3???3?2? 当t??时?
dx1331?3?22 7? 写出下列曲线在所给参数值相应的点处的切线方程和法线方程?
?x?sint (1) ?? 在t??处?
4?y?cos2t?x?3at?1?t2 (2) ?2? 在t=2处? 3at?y??1?t2dyy? 解 (1)?t??2sin2t??
dxxt?cost?)?2sin(2?dy4??2??22? x?2? y?0??? 当t??时? 00?2dx42cos42所求切线方程为?
y??22(x?2)? 即22x?y?2?0?
2所求法线方程为
y??1(x?2)? 即2x?4y?1?0?
2?226at(1?t2)?3at2?2t6at? (2)yt???(1?t2)2(1?t2)23a(1?t2)?3at?2t3a?3at2 xt??? ?(1?t2)2(1?t2)2dyyt???6at2?2t2?
dxxt?3a?3at1?tdy2?24??? 当t?2时? ? x0?6a? y0?12a? 2dx1?2355所求切线方程为?
y?12a??4(x?6a)? 即4x?3y?12a?0?
535所求法线方程为
y?12a?3(x?6a)? 即3x?4y?6a?0?
545d2y 8? 求下列参数方程所确定的函数的二阶导数2?
dx2??x?t (1) ?2?
??y?1?t.?x?acost (2) ??
y?bsint?