100 Chapter 4Differentiation
EXAMPLE 4.5
From the properties of the logarithm,
0 Exercises 16, 17
4.5 Differentiation from first principles
A function is said to be differentiableat a point if the limit in equation (4.8),
(4.8)
exists and is unique. A necessary condition for this to be true is that the function be
continuous at the point, but not all continuous functions are everywhere differentiable.
For example, the function
is continuous at all values of xbut Figure 4.7 shows that its
slope changes abruptly atx 1 = 10 , from value –1 whenx 1 < 10 to
value +1 forx 1 ≥ 10. The function has a cuspatx 1 = 10 and the derivative is not defined
by equation (4.8).
In general, a function is differentiable if it is continuous and ‘smooth’, with no
essential discontinuities or cusps. For any such function, the taking of the limit in
equation (4.8) is called differentiation from first principles, and was demonstrated
in Examples 4.1 and 4.2 for the general quadratic. All functions can be differentiated
in this way. Lety 1 = 1 f(x)be a function of x, and let the function change from y
to y 1 + 1 ∆y 1 = 1 f(x 1 + 1 ∆x)when the variable changes fromxto x 1 + 1 ∆x. The steps for
differentiating from first principles are:
(1) subtract yfromy 1 + 1 ∆yto obtain ∆yas a function of xand ∆x, and simplify as
much as possible,
(2) divide both sides of the equation by ∆x,
(3) find the limit of as∆x 1 → 10 ; this gives the required derivative.
0 Exercises 18, 19
dy
dx
∆
∆
y
x
fx x
xx
xx
() | | ==
≥
−<
if
if
0
0
dy
dx
y
x
fx x fx
xx
=
=
+−
→→
lim lim
()()
∆∆
∆
∆
∆
∆
00
xx
ln( 23 ) ln( 2 ) ln ln
23
2
xx 2
x
x
+− − = x
−
→→as ∞
lim ln( ) ln( ) ln
x
xx
→
+− −
=
∞
23 2 2
f(x)
x
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Figure 4.7