Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 343

ALTERNATIVE NOTATION

If we are given a function f(x) then f^0 (x) represents the derivative function.

If we are givenyin terms ofxtheny^0 or

dy

dx

are commonly used to represent the derivative.

dy

dx

reads “deeyby deex” or “the derivative ofywith respect tox”.

dy

dx

isnot a fraction. However, the notation

dy

dx

is a result

of taking the limit of a fraction. If we replacehby ±x and

f(x+h)¡f(x) by ±y, then

f^0 (x) = lim

h! 0

f(x+h)¡f(x)

h

becomes

f^0 (x) = lim

±x! 0

±y

±x

=

dy

dx

.

THE DERIVATIVE WHEN x=a

The gradient of the tangent to y=f(x) at the point where x=a is denoted f^0 (a), where

f^0 (a) = lim

h! 0

f(a+h)¡f(a)

h

Example 4 Self Tutor

Use the first principles formula f^0 (a) = lim

h! 0

f(a+h)¡f(a)

h

to find the

instantaneous rate of change in f(x)=x^2 +2x at the point where x=5.

f(5) = 5^2 + 2(5) = 35

) f^0 (5) = lim

h! 0

f(5 +h)¡f(5)

h

= lim

h! 0

(5 +h)^2 + 2(5 +h)¡ 35

h

) f^0 (5) = lim

h! 0

25 + 10h+h^2 +10+2h¡ 35

h

= lim

h! 0

h^2 +12h

h

= lim

h! 0

h(h+ 12)

h

fas h 6 =0g

=12

) the instantaneous rate of change in f(x) at x=5is 12.

1

y

x x+ x± x

y

y+ y±

y = f(x)

O

±y

±x

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_13\343CamAdd_13.cdr Tuesday, 7 January 2014 2:36:22 PM BRIAN