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

11.2 Rules of differentiation 389
The quotient rulestates: when
y(x) = u(x)
v(x)
then
( ) ( ) du dv
v −u
y ′ dx dx
=
= vu′ −uv ′
v 2 v 2
Example11.2 Findy ′ given
(a)y= sinx
x
t2
(b)y=
2t+1
e2t
(c)y=
t 2 +1
Solution (a)y= sinx = u x v .Sou=sinx,v=xandu′ =cosx,v ′ = 1. Using the quotient rule
the derivative ofyisfound:
(b)y=
y ′ = xcosx−sinx
x 2
t2
2t+1 = u v .Sou=t2 ,v=2t+1andu ′ =2t,v ′ = 2.Hence,
y ′ = (2t +1)2t − (t2 )(2) 2t(t +1)
=
(2t +1) 2 (2t +1) 2
(c)y=
e2t
t 2 +1 .Sou=e2t ,v=t 2 +1andu ′ = 2e 2t , v ′ = 2t. Application of the
quotient ruleyields
y ′ = (t2 +1)2e 2t −e 2t 2t
= 2e2t (t 2 −t+1)
(t 2 +1) 2 (t 2 +1) 2
11.2.3 Thechainrule
Thisrulehelpsustodifferentiatecomplicatedfunctions,whereasubstitutioncanbeused
to simplify the function. Supposey = y(z) andz = z(x). Thenymay be considered as
afunctionofx.Forexample,ify =z 3 −zandz = sin3x,theny = (sin3x) 3 −sin(3x).
Suppose weseekthe derivative, dy . Note that the derivative w.r.t.xissought.
dx
Thechain rule states:
dy
dx = dy
dz × dz
dx