96 Chapter 4Differentiation
As Q moves through Q′towards P, the gradient of the line PQ approaches the
gradient of the tangent at P. We express this as
(4.6)
(read as ‘the limit as Q goes to P’). At the same time, the magnitude of∆xgoes to zero,
∆x 1 → 10 asQ 1 → 1 P, and the limit can be expressed as
1(4.7)
EXAMPLE 4.2For the quadratic functiony 1 = 1 ax
21 + 1 bx 1 + 1 cin Example 4.1,
The limit is a function of x, and it gives the gradient or slope of the curve at each value
of x(each point on the curve).
0 Exercises 2, 3
The process of taking the limit in (4.7) is called differentiation. In the differential
calculus, the limit is denoted by the symbol :
(4.8)
(read as ‘dy by dx’).
2It is called the differential coefficientof the function or, simply,
the derivativeof the function. We note that the symbol does notmean the quantity
‘dy’ divided by the quantity ‘dx’; the symbol represents the limit, and the taking of the
limit as given by the right side of (4.8).
dy
dx
dy
dx
y
x
fx x fx
x x
=
=
+−
→ →
lim lim
()()
∆∆
∆
∆
∆
0 ∆ 0
xx
dy
dx
∆
∆
∆
∆
∆ ∆
y
x
ax b a x
y
x
ax
x
=++
=
→
( 22 ) lim
0
so that ++b
gradient atx
y
x
x
=
→
lim
∆
∆
∆
0
gradient at P
QP
=
→
lim
∆
∆
y
x
1This method of finding the tangent at a point on a curve is essentially that given by Fermat in his Method of
finding maxima and minimain about 1630. This work marks the beginning of the differential calculus. A method
similar to Fermat’s but involving quantities equivalent to ∆xand ∆ywas described by Barrow in Lectiones
geometriae, published in 1670. In these lectures Isaac Barrow (1630–1677), theologian and professor of geometry
at Cambridge, gave a ‘state of the art’ account of infinitesimal methods. The formulation of the method of tangents
was included ‘on the advice of a friend’, Newton, who succeeded him in his chair when Barrow became chaplain
to Charles II in 1669.
2The notation is derived from Leibniz’s formulation of the calculus. Gottfried Wilhelm Leibniz (1646–1716),
philosopher, diplomat and mathematician, discovered his form of the calculus in the years 1672–1676 whilst on
diplomatic service in Paris, where he came under the influence of the physicist and mathematician Christiaan
Huygens, inventor of the pendulum clock. His first account of the differential calculus was the Nova methodus
pro maximis et minimis, itemque tangentibus(A new method for maxima and minima, and also for tangents),
published in 1684.