MATH 110 BUSINESS CALCULUS
CHAPTER 3 INTRODUCTION TO THE DERIVATIVE
Section 3.5 Derivatives: Numerical and Graphical
Viewpoints
The
instantaneous rate of change
of f(x) at x = a is
defined as
0
(
)
( )
( )
lim
h
f a
h
f a
f
a
h
→
+
−
′
=
[We say “
f prime of a equals the limit as h approaches 0,
of the ratio
(
)
( )
f a
h
f a
h
+
−
”]
( )
f
a
′
is also called the
derivative
of f(x) at x = a.
Differentiation
is the process of finding the derivative.
Leibniz d Notation:
0
0
(
)
( )
( )
lim
lim
h
h
f x
h
f x
f
df
f
x
h
x
dx
→
→
+
−
Δ
′
=
=
=
Δ
Thus,
(
) can be written as
in Leibniz Notation
x
a
df
f
a
dx
=
′
The vertical line indicates that the derivative is being
evaluated at x = a.

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