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

11.2 Rules of differentiation 387
The product rulestates: if
then
y(x) =u(x)v(x)
dy
dx = du
dx v +udv dx =u′ v+uv ′
To apply this rule one of the functions in the product must be chosen to beu, and the
other, v. Beforewe can apply the ruleweneed tocalculateu ′ and v ′ .
Example11.1 Findy ′ given
(a) y=xsinx
(b)y=t 2 e t
Solution (a) y = xsinx = uv. Chooseu = x and v = sinx. Thenu ′ = 1, v ′ = cosx. Applying
the productruletoyyields
y ′ =sinx+xcosx
(b)y=t 2 e t =uv.Chooseu =t 2 andv =e t .Thenu ′ =2tandv ′ = e t . Applying the
product ruletoyyields
y ′ =2te t +t 2 e t
Engineeringapplication11.1
Dampedsinusoidalsignal
Acommonfunctionfoundinengineeringisthedampedsinusoidalsignal.Thisconsists
of a negative exponential function multiplied by a sinusoid. A typical example
is
f (t) = e −0.1t cost
The graph of this function is shown in Figure 11.1. This function approximates the
wayacarbodyreactswhenthecardrivesoveralargebumpintheroad.Fortunately,
thecarshockabsorbersensurethattheoscillationsreduceinamplitudequitequickly.
When sketching such a function it can be useful to think of the exponential term,
and its mirror image around the time axis, providing an envelope that contains the
signal. When values of the sinusoid are 1 then the signal touches the positive part
of the envelope and when values of the sinusoid are −1 then the signal touches the
negative partofthe envelope.
Therateofchangeofthissignalwithrespecttotimecanbefoundbydifferentiating
f (t)usingtheproductrule.Todosowenotethat f (t)isaproductofu(t) = e −0.1t
and v(t) = cost.Recastingtheformulafordifferentiatingaproductintermsoftwe
have
f(t) =u(t)v(t)
➔