(ii)a^3 xax
a^2 x=a^3 x+x−^2 x=a^2 x(iii)(ax)^3 a−^2 x
(a^4 )x=a^3 xa−^2 x
a^4 x=a−^3 x(iv)ax
2
a−^2 x
a(x−^1 )^2=ax(^2) − 2 x−(x− 1 ) 2
=a−^1 using a bit of elementary algebra
(v)
2 x 16 −^3 x
42 x 8 −x+^1
= 2 x 2 −^12 x 2 −^4 x 23 x−^3 = 2 −^12 x−^3
4.2.3 The natural exponential functionex
➤
119 136➤
The most commonly used exponential function – usually referred to asthe exponen-
tial function–isex,whereeis a number whose value to 16 decimal places (!) is
e
2 .7182818284590452. eis called the base of natural logarithms. It is an irra-
tional number. Likeπ, it cannot be expressed as a fraction, or as a terminating or
repeating decimal. Its decimal part goes on indefinitely. There are a number of equiv-
alent definitions we could give forex, but it has to be said that none of them is easy to
appreciate at the elementary level. Below we will study in detail perhaps the most natural
derivation of an expression forex, by considering compound interest. For now, accept
the following definitions as a general introduction to the properties of the exponential
function.
We will see later that the particular exponential functionexcan be defined by its infinite
series
ex= 1 +x+
x^2
2!
- x^3
3!
+···
Using this form, if you know some elementary differentiation, then you can discover for
yourself why the exponential function is so important:
Problem 1
Differentiate the series forexterm by term – what do you get? You may
assume
d
dx
.xn/=nxn−^1
You should find you get the same series – i.e.
dex
dx
=ex
So the derivative of the exponential function is the function itself. This explains the special
role played byexand its importance in, for example, differential equations (Chapter 15).
There are many situations where rate of change is proportional to amount present – e.g.
bacterial growth, radioactive decay (e−xis the relevant function in the latter case).