1Differentiation

IA Differential Equations


1.1 Differentiation
Definition (Derivative of function). The derivative of a function
f
(
x
) with
respect to x, interpreted as the rate of change of f(x) with x, is
df
dx
= lim
h→0
f(x + h) − f(x)
h
.
A function
f
(
x
) is differentiable at
x
if the limit exists (i.e. the left-hand and
right-hand limits are equal).
Example.
f
(
x
) =
|x|
is not differentiable at
x
= 0 as
lim
h→0
+
|h|−|0|
h
= 1 and
lim
h→0
−
|h|−|0|
h
= −1.
Notation. We write
df
dx
=
f
0
(
x
) =
d
dx
f
(
x
). Also,
d
dx
d
dx
f(x)
=
d
2
dx
2
f
(
x
) =
f
00
(x).
Note that the notation
f
0
represents the derivative with respect to the
argument. For example, f
0
(2x) =
df
d(2x)