Page 8 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation
Derivative of e x

x
Consider the exponential function f(x) = a in Figure 2.3 where a is a positive constant. For any value of a,
0
a = 1. Hence, the curve passes through the point A(0, 1).
The gradient at point A is given by

f(0) = lim f(0 + x) – f(0)
x → 0 x
= lim f(x) – f(0) y y = a x
x → 0 x
x
= lim a – a 0
x → 0 x 2
x
= lim a – 1 A(0, 1)
x → 0 x
x
0
For f(x) = a x
d a x + x – a x Figure 2.3
x
(a ) = lim
dx x → 0 x
x
= lim a x 1 a – 1 2
x → 0 x
x
a – 1
= a x lim
x → 0 x
x
= a f(0) f(0) = x → 0 a – 1
x
lim
x
d
x
Hence, (a ) = a f(0)
x
dx
Now there must be a value of a for which f(0) = 1, i.e. the gradient of the graph at (0, 1) is 1. We call this
value e.
x
Thus d (e ) = e x
dx
e is an irrational number. e ≈ 2.718.
e is known as the exponential function.
x

Derivative of ln x
Let y = f(x), where f(x) is any function of x,
dy y
lim
hence = x → 01 2
dx x
= lim 1
x → 0 1 2
x
y

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02 STPM Math(T) T2.indd 21 02/11/2018 12:43 PM