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02.01.1
Chapter 02.02
Differentiation of Continuous Functions
After reading this chapter, you should be able to:
1.
derive formulas for approximating the first derivative of a function,
2.
derive formulas for approximating derivatives from Taylor series,
3.
derive finite difference approximations for higher order derivatives, and
4.
use the developed formulas in examples to find derivatives of a function.
The derivative of a function at
x
is defined as
( )
( ) ( )
x
x
f
x
x
f
x
f
x
∆
−
∆
+
=
′
→
∆
0
lim
To be able to find a derivative numerically, one could make
x
∆
finite to give,
( )
( ) ( )
x
x
f
x
x
f
x
f
∆
−
∆
+
≈
′
.
Knowing the value of
x
at which you want to find the derivative of
( )
x
f
, we choose a value
of
x
∆
to find the value of
( )
x
f
′
.
To estimate the value of
( )
x
f
′
, three such approximations
are suggested as follows.
Forward Difference Approximation of the First Derivative
From differential calculus, we know
( )
( ) ( )
x
x
f
x
x
f
x
f
x
∆
−
∆
+
=
′
→
∆
0
lim
For a finite
x
∆
,
( )
( ) ( )
x
x
f
x
x
f
x
f
∆
−
∆
+
≈
′
The above is the forward divided difference approximation of the first derivative.
It is called
forward because you are taking a point ahead of
x
.
To find the value of
( )
x
f
′
at
i
x
x
=
, we
may choose another point
x
∆
ahead as
1
+
=
i
x
x
.
This gives
( )
( ) ( )
x
x
f
x
f
x
f
i
i
i
∆
−
≈
′
+
1

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