082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

392 Chapter 11 Techniques of differentiation
2 Usethe quotient ruleto findthe derivatives ofthe
following:
cosx tant
(a) (b)
sinx lnt
(c)
(e)
(g)
e 2t
t 3 +1
(d)
x 2 +x+1
1+e x (f)
1+e t
1+e 2t
3x 2 +2x−9
x 3 +1
sinh2t
cosh3t
3 Usethe chainruleto differentiate the following:
(a) (t 3 +1) 100 (b) sin 3 (3t +2)
(c) ln(x 2 +1) (d) (2t +1) 1/2
(e) 3 √ cos(2x −1)
(g) (at +b) n ,aandbconstants
(f)
1
t +1
4 Differentiate eachofthe followingfunctions:
(a) y=5sinx
(c) y = 5e sinx
(e) y = (t 3 +4t) 15
(g)y=
sinx
4cosx+1
(b) y=5e x sinx
(d)y= 5sinx
e −x
(f) y = 7e −3t2
5 Forwhichvalues oft is the derivative ofy(t) = e −t t 2
zero?
6 Findtherateofchangeofyatthespecifiedvaluesoft.
(a) y = sin 1 t = 1
t
(b) y=(t 2 −1) 17 t=1
(c) y =sinh(t 2 ) t =2
(d) y= 1+t+t2
1 −t
(e) y=
et
tsint
t = 1
t = 2
7 Findthe equationofthe tangent to
8 Differentiate
y(x) = e 3x (1 −x) atthepoint (0,1)
(a) y=lnx
(b) y=ln2x
(c) y = lnkx,k constant
(d) y =ln(1+t)
(e) y =ln(3+4t)
(f) y =ln(5 +7sinx)
Solutions
1 (a) cos 2 x −sin 2 1
x (b)
t tant+lntsec2 t
e x (−x 2 +x)+2x+1
(e)
( )
(e x +1) 2
√x
(c) e 2t (2t 3 +3t 2 +2) (d) e x 1 +
2 √ 2cosh2tcosh3t −3sinh2tsinh3t
x
(f)
(cosh3t) 2
(e) e t (2cos 2 t +sintcost −1)
−e 3t −2e 2t +e t
(f) 3[2cosh2tcosh3t +3sinh2tsinh3t]
(g)
(e 2t +1) 2
(g) (1+sint)sec 2 t +sint
3 (a) 300t 2 (t 3 +1) 99
(h) 4[cosh(t +1)cosh(1 −t)
−sinh(t +1)sinh(1 −t)]
(b) 9sin 2 (3t +2)cos(3t +2)
2 (a) −cosec 2 x
2x
(c)
x 2 +1
lntsec 2 t − (tant)/t
(b)
(lnt) 2
(d) (2t +1) −1/2
e 2t (2t 3 −3t 2 −3sin(2x −1)
+2)
(e) √
(c)
cos(2x −1)
(t 3 +1) 2
−3x 4 −4x 3 +27x 2 (f) −(t +1) −2
+6x +2
(d)
(x 3 +1) 2 (g) an(at +b) n−1