3.9Differentials
Let y = f(x) represent a function that is differentiable inan open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differentialof y (denoted by dy) is given by
dy = f’(x) dx
x
y
dx
dyx Δ
ΔΔ 0lim
→=
x xx Δ+
xdx Δ=
dy
yΔ
)()( xfxxfy −+= ΔΔ
Comparing dyandyΔLet y = x2. Find dy when x = 1 and dx = 0.01. Comparethis value to when x = 1 and = 0.01.yΔ xΔ
dy = f’(x) dx dy = 2x dx
dy = 2(1)(.01) = .02
)()( xfxxfy −+= ΔΔ = f(1.01) – f(1)
= 1.012 – 12 = .0201
(1, 1)
01.== xdx Δ
0201.0=yΔdy = 0.02
Estimation of error.
The radius of a ball bearing is measured to be .7 inch. If themeasurement is correct to within .01 inch, estimate thepropagated error in the Volume of the ball bearing.
r = .7
r = .7 and 01.01. ≤≤− rΔ
3
3
4rV π=
drrdVV 24πΔ ==
( ) ( )01.07.04 2 ±= π
06158.0±≈ propagated error
relative error is
( ) %29.4100 ≈V
dV( )0429.
7.3406158.
3±=±
π Percentage error
Finding differentials
Function Derivative Differential
y = x2 xdx
dy2= dy = 2x dx
y = 2sin x xdx
dycos2= dy = 2cos x dx
y = x cos x )1(cos)sin( xxxdx
dy+−=
( )dxxxxdy cossin +−=
y = sin 3x xdx
dy3cos3= dxxdy 3cos3=
xxf =)(101Find
)()( xfxxfy −+= ΔΔ
yxfxxf ΔΔ +=+ )()(
dyxfxxf +=Δ+ )()( dyy ≈Δand dy = f’(x) dx
dxxfxfxxf )(')()( +=+∴ Δ
Let x = 100and 1==dxxΔ
xxf
2
1)(' =
( )11002
1100 + 05.10
20
110 =+=
05.10101 ≈∴
=+ )1100(f