4.5 Differentiation from first principles 101
EXAMPLE 4.6Find from first principles for.
(1)
(2) (3)
0 Exercise 20
EXAMPLE 4.7Differentiate.
(1)
(making use of the relation (a 1 − 1 b)(a 1 + 1 b) 1 = 1 a
21 − 1 b
2).
(2) (3)
0 Exercise 21
EXAMPLE 4.8Differentiatee
x.
(1)y 1 = 1 e
x, y 1 + 1 ∆y 1 = 1 e
x+∆x1 = 1 e
x1 × 1 e
∆xFrom the definition of the exponential function as an infinite series, equation
(3.31),
Therefore,
=+eex+ + +
xx
xx(
() ()
∆ )
∆∆
2326
yye x
xx
x+=×++ + +∆∆
∆∆
(
() ()
1 )
26
23ex
xx
∆x∆
∆∆
=+ + 1 + +
23
23()
!
()
!
dy
dx
y
x
xx
=
=
→
lim
∆
∆
0 ∆
1
2
∆
∆
∆y
x
xx x=
++
1
=
+−
++
=
++
()xxx
xx x
x
xx x
∆
∆
∆
∆
∆∆
∆∆
∆
yxxx
xxxxx
xx x
=+− =
+− × + +
++
()( )
y x yy xx=+=+, ∆∆
x
dy
dx
y
x
x
x
=
∆
∆
=−
→
lim
∆ 0
1
2∆
∆∆
y
xxx x
=
−
1
()
∆∆
∆
∆
∆
yfx x fx
xxx
x
xx x
=+− =
−=
−
()()
()
11
yfx
x
yyfxx
xx
==,+=+=
() ( )
11
∆∆
∆
y
x
=
dy 1
dx