APPENDIX 1
Differentiation from
First Principles
We hinted in Section 4.1 that there was a formal way of actually proving the formulae for
derivatives. This is known as ‘differentiation from Frst principles’ and we begin by illustrating
the basic idea using a simple example. ±igure A1.1 shows the graph of the square function
f
(
x
)
=
x
2
near
x
=
3
.
The slope of the chord joining points A and B is
=
=
==
6
+Δ
x
Now, as we pointed out in Section 4.1, the slope of the tangent at
x
=
3
is the limit of the slope
of the chords as the width,
Δ
x
, gets smaller and smaller. In this case
slope of tangent
=
lim
Δ
x
→
0
(6
x
)
=
6
In other words, the derivative of
f
(
x
)
=
x
2
at
x
=
3
is 6 (which agrees with
f
′
(
x
)
=
2
x
evaluated at
x
=
3
). Notice that this proof is not restricted to positive values of
Δ
x
. The chords in ±igure A1.1
could equally well have been drawn to the left of
x
=
3
. In both cases the slope of the chords
approaches that of the tangent at
x
=
3
as the width of the interval shrinks.
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 Spring '11
 efeeewewf
 Calculus, Slope, Cellular differentiation, Δx

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