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

372 Chapter 10 Differentiation
Solutions
1 See FigureS.22.
(a) No derivative exists for t = 1
y
(b) No derivative exists for t = nπ
|sin t|
(c) Derivative exists for all values of t
e t
1
t
–2π–π
0 π 2π3π
t
0
t
(d) No derivative exists for t = 0
1
–
t
FigureS.22
0 t
(e) No derivative exists for t = 0
u(t)
1
t
(f) No derivative exists for t = 0, although
the function is continuous here
f(t)
t
10.7 COMMONDERIVATIVES
Itistimeconsumingtofindthederivative ofy(x) usingthe‘shrinkinginterval’method
(often referred to as differentiation from first principles). Consequently the derivatives
of commonly used functions are listed for reference in Table 10.1. It will be helpful
to memorize the most common derivatives. Note that a, b and n are constants. In
the trigonometric functions, the quantity ax +b, being an angle, must be measured in
radians.
A shorter table of the more common derivatives is given on the inside back cover of
this bookforeasyreference.
Example10.11 UseTable 10.1 tofindy ′ when
(a) y = e −7x
(b)y=x 5
(c) y = tan(3x −2)
(d) y = sin(ωx + φ)
(e)y= √ 1 x
(f)y= 1 x 5
(g) y = cosh −1 5x