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

378 Chapter 10 Differentiation
(d) The linearityproperties allowus todifferentiate each termindividually. If
y = sin2x −cos5x
then
dy
dx = d dx (sin2x) − d dx (cos5x)
=2cos2x+5sin5x
(e) Ify = 3lnx,then
dy
dx = d dx (3lnx)=3d dx (lnx)
= 3 x
using linearity
using Table 10.1
(f) Ify = ln2x, then
y = ln2 +lnx
usinglaws of logarithms
and so
dy
dx = 0 + 1 x = 1 x
(g) Each termisdifferentiated:
y ′ =6x+7−0=6x+7
since ln2 isconstant
Engineeringapplication10.4
Dynamicresistanceofasemiconductordiode
RecallfromEngineeringapplication2.9thatasemiconductordiodecanbemodelled
by the equation
I =I S
(e qV
nkT −1)
where V is the applied voltage, I is the diode current, I S
is the reverse saturation
current, k is the Boltzmann constant, q is the charge on the electron, and T is the
temperature of the diode junction in Kelvin. The parameter n is the ideality factor
whichisusuallyconsideredtobeaconstantforagiventypeofsemiconductorjunction,
dependent primarily on the design and manufacture ofthe diode.
Examiningthediodeelectricalcharacteristics,showninFigure2.31,wenotethat
there are two main electrical regimes. The first occurs when the applied voltage,V,
isnegativeandthecurrent,I,islimitedtoanegativesaturationcurrent,I s
,apartfrom
a small region around the origin. The second occurs when the applied voltage,V, is
positive and the current,I, takes on the shape of an exponential curve, apart from a
smallregion around the origin.
Wewillconcentrateonthesecondcaseandfurthermorewewillassumethate qV
nkT
is very much greater than 1. This is written concisely as e qV
nkT >> 1. This is a good
approximationapartfromthesmallregionclosetotheorigin.Theoriginalexpression
forIcan then be writtenas
I≈I S
e qV
nkT
where the symbol ≈means ‘is approximately equal to’.