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

10.8 Differentiation as a linear operator 375
Solutions
1 (a) 2t (b) 9t 8
(c) −3t −4 (d) 1
(e) −t −2 (f) −2t −3
(g) 3e 3t (h) −3e −3t
(i) −5e −5t (j) 0.5t −1/2
(k) 2cos(2t +3) (l) sin(4 −t)
(m) 0.5sec 2 (t/2 +1)
(n) −3cosec(3t +7)cot(3t +7)
(o) cosec 2 (1 −t)
(p) 2sec(2t − π)tan(2t − π)
1
(q) √ (r) 0
1−(t+π) 2
2
4
(s) −
1 + (−2t −1) 2 (t) −√ 1−(4t−3) 2
(u) 6sech 2 6t
( )
(v) 2sinh(2t +5)
t +3
(w) 0.5cosh
2
(x) sech(−t)tanh(−t)
(y) − 2 3 cosech2 (
2t
3 − 1 2
)
(z)
1
√
(t+3) 2 −1
2 (a) −0.5x −3/2 2
(b)
3 e2x/3
(c) −0.5e −x/2 (d) 1/x
(e) − 2 3 cosec (
2x−1
3
)
cot
(
2x−1
3
π
(f)
1+(πx+3) 2 (g) 2sech 2 (2x +1)
3
(h) −√ (i) −ωcosec 2 (ωx + π)
9x 2 +1
(j) −5cosec(5x +3)cot(5x +3)
(k) −3sin3x
(l) 3sec3xtan3x (m) 2sec 2 (2x + π)
( ) ( )
x −1 x −1
(n) −0.5cosech coth
2 2
(o)
2
[ ( ) 2 ]
2x+3
7 1 −
7
)
10.8 DIFFERENTIATIONASALINEAROPERATOR
Inmathematicallanguagedifferentiationisalinearoperator.Thismeansthatifwewish
to differentiate the sum of two functions we can differentiate each function separately
and thensimplyadd the two derivatives, thatis
derivative of (f +g) = derivative of f +derivative ofg
This isexpressed mathematically as
d
dx (f+g)=df dx + dg
dx
We can regard d as the operation of differentiation being applied to the expression
dx
whichfollows it.Thepropertiesofalinear operator alsomake thehandling ofconstant
factorseasy.Todifferentiatekf,wherekisaconstant,wetakektimesthederivativeof
f,thatis
derivative of (kf) =k ×derivative of f
Mathematically, wewould state:
d
dx (kf)=kdf dx