The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 109 Friday, March 12, 2004 12:39 PM

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Principles of Calculus 109

The first term is just a and as per Rule 1 the derivative is zero. The

derivative of b 1 x by Rule 2 is b 1. For each term bnxn by Rule 2 the

derivative is nbnxn–1. Thus, the derivative of

b 2 x^2 is 

2 b 2 x^1

b 3 x^3 is 

3 b 3 x^2

b 4 x^4 is 

4 b 4 x^3

etc.

Therefore, the derivative of y is

dy 2 n – 1

-------= b 1 + 2 b 2 x

1

+ 3 b 3 x + 4 b 4 x

3

+ ...+ nbnx

dx

There is a special rule for a composite function. Consider a compos-

ite function: h(x) = f[g(x)]. Provided that h and g are differentiable at

the point x and that f is derivable at the point s = g(x), then the follow-

ing rule, called the chain rule, applies:

h′()x = f ′(gx())g′()x

hx()= fgx( ())

dh df   dg

-------=  

dx dg dx  

Exhibit 4.6 shows the sum rule, product rule, quotient rule, and

chain rule for calculating derivatives in both standard and infinitesimal

notation. In Exhibit 4.6 it is assumed that a,b are real constants (i.e.,

fixed real numbers), that f, g, and h are functions defined in the same

domain, and that all functions are differentiable at the point x. Exhibit

4.7 lists (without proof) a number of commonly used derivatives.

Given a function f(x), its derivative f ′(x) represents its instanta-

neous rate of change. The logarithmic derivative

d f ′()x

-------ln Px()= ------------

dx fx()

for all x such that P(x) ≠ 0, represents the instantaneous percentage

change. In finance, the function p = p(t) represents prices; its logarith-

mic derivative represents the instantaneous returns.