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

376 Chapter 10 Differentiation
Table 10.1 together with these two linearity properties allow us to differentiate some
quite complicated functions.
Example10.13 Differentiate
(a) 3x 2 (b) 9x (c) 7 (d) 3x 2 +9x +7
Solution (a) Lety = 3x 2 , then
dy
dx = d dx (3x2 )
= 3 d dx (x2 ) using linearity
= 3(2x) fromthe table
= 6x
(b) Lety = 9x, then
dy
dx = d dx (9x)
= 9 d (x) using linearity
dx
= 9
(c) Lety = 7,theny ′ = 0.
(d) Lety=3x 2 +9x+7
dy
dx = d dx (3x2 +9x +7)
dy
dx = 3 d dx (x2 )+9 d dx (x) + d (7) usinglinearity
dx
=6x+9
Engineeringapplication10.3
Fluidflowintoatank
If fluid is being poured into a tank at a rate of q m 3 s −1 , then this will result in an
increase in volume,V, of fluid in the tank. The arrangement is illustrated in Figure
10.16. Therate ofincrease involume, dV m 3 s −1 , isgiven by
dt
dV
=q
dt
This relationship follows from the principle of conservation of mass. If q is large,
then dV is large, which corresponds to the fluid volume in the tank increasing at a
dt