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

10.8 Differentiation as a linear operator 377
fast rate. Consequently, the height of the fluid,h, also increases at a fast rate. If the
cross-sectional area ofthe tank,A, isconstant, thenV =Ah. Therefore,
dV
= d (Ah) =Adh
dt dt dt
because differentiation isalinear operator. So,
A dh
dt
=q
q
h
V
Figure10.16
A closedtankcontaining avolume offluidV.
Example10.14 UseTable 10.1 and the linearityproperties ofdifferentiation tofindy ′ where
(a) y=3e 2x
(b) y = 1/x
(c) y=3sin4x
(d) y = sin2x −cos5x
(e) y=3lnx
(f) y=ln2x
(g) y=3x 2 +7x−5
Solution (a) Ify = 3e 2x , then
dy
dx = d dx (3e2x ) = 3 d dx (e2x ) usinglinearity
= 3(2e 2x ) usingTable 10.1
=6e 2x
(b) Ify =x −1 ,then
y ′ = −1x −2 from Table 10.1
= − 1 x 2
(c) Ify = 3sin4x, then
dy
dx = d dx (3sin4x) =3 d dx (sin4x) usinglinearity
= 3(4cos4x) usingTable 10.1
= 12cos4x