The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 108 Friday, March 12, 2004 12:39 PM

108 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 4.5 Geometric Interpretation of a Derivative

d

■ Rule 1: -------()c = 0 , where c is a real constant.

dx

d

■ Rule 2: -------(bx

n

)= nbx

n – 1

, where b is a real constant.

dx

d d d

■ Rule 3: -------(af x()+ bg x()) = a-------fx()+ b-------gx(), where a and b are

dx dx dx

real constants.

Rule 3 is called the rule of termwise differentiation and shows that dif-

ferentiation is a linear operation.

Let’s apply the basic rules to the following function:

y = a + b k

1 x + b 2 x

(^2) + b
3 x
(^3) + ... + b
kx
where a, b 1 , b 2 , b 3 , ..., bk are the constants.