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

400 Chapter 11 Techniques of differentiation
2 (a) e ydy
dx
(c) −2sin2y dy
dx
1
(e)
2 y−1/2dy dx
(
(g) −sin(x +y)
(h)
3 (a)
(b)
1 dy
y dx
1+9x 2
4y−1
(4x √ x−1) √ y
(1−6y 2√ y) √ x
(b) cosy dy
dx
(d) −3e −3ydy
dx
(f) 1+ dy
dx
)
1 + dy
dx
1 dy
(i)
y dx
2
(c)
3 (ex√ 2x+3y−1)
(
1
(d) y
2(1 +x) +1− 2 )
x
(e)
(f)
(g)
3x 2 +3y 2 −2y 4
2xy(4y 2 −3)
cos(x +y)
1 −cos(x +y)
2(x 2 +y 2 −x)
3x 2 +3y 2 +2y
5 (a) (4x 3 +x 4 )e x
( )
(b) −e −x 1
x 2 + 1 x
( )
(c) t 3 (1+t) 9 3
t + 9
1 +t
(d) e x sinx(1 +cotx)
( )
(e) x 7 sin 4 7
x
x +4cotx
(
)
4
6 (a)z
t − 6
1 −t + 4
2 +t
(
)
6x
(b) y
1+x 2 +7− 6
2 +x
(
)
3
(c) x
1 +t + 4
2 +t + 5
3 +t
(
)
(d) y 4cott −
8t
2−t 2 − 6et
1+e t
( )
(e) x 3 e x 3
sinx
x +1+cotx
7 −0.1387
8 (a) 192 (b) −2 (c) − 1 3
(h)
e (x/2−2y) x(x +4)
2(1 +2y)
4 (a) 3t/2 (b)
(d)
−3
2sin2t
e t
cost
(c)
(e) −2e 2t t 2 /3 (f)
( ) 2
t
1 +t
e t −e −t
e t +e −t
(d)
dx
dy = −3t4 ;whent =2, dx
dy = −48
9 y = −4x+6 ,y = 4x+24
5 5
10 y ′ = 6x
1−3y 2
11.4 HIGHERDERIVATIVES
The derivative, y ′ , of a function y(x) is more correctly called the first derivative of y
w.r.t.x.Sincey ′ itselfisafunctionofx,thenitisoftenpossibletodifferentiatethistoo.
Thederivative ofy ′ iscalled the secondderivative ofy:
secondderivative ofy = d dx
( ) dy
dx
which iswrittenas d2 y
dx 2 or more compactly asy′′ .